On Multiparameter Post-Quantum Fractional Quadrature Inequalities with Simulation

Abstract

This paper introduces a comprehensive class of multiparameter post-quantum fractional quadrature inequalities, unifying classical error bounds within the setting of the post-quantum Riemann–Liouville fractional integral. By incorporating multiple parameters, we derive a flexible family of inequalities that generalize well-known quadrature rules such as the Boole-type, Bullen–Simpson-type, Maclaurin-type, corrected Euler–Maclaurin-type, ${\displaystyle \frac{3}{8}}$-Simpson-type, and companion Ostrowski-type estimates. Under assumptions of s-convexity, log-convexity, power mean inequality, and Holder inequality, we establish novel error bounds. Our results provide a unified framework for designing and analyzing post-quantum fractional quadrature inequalities. Applications to special means and numerical and graphic examples are presented to illustrate the applicability and generality of the derived inequalities. This work lays a theoretical foundation for the development of post-quantum fractional quadrature inequalities and offers new tools for error estimation in post-quantum fractional-order models arising in applied sciences and engineering.

1. Introduction

The classical calculus of Newton and Leibniz, built upon the foundational concept of the limit, has long been the principal language for modeling continuous change. However, many contemporary problems in mathematics and physics, from quantum mechanical systems to digital signal processing, inherently involve discrete or quantized structures. This necessity inspired the development of quantum calculus (or q-calculus), a transformative framework that redefines differentiation and integration without recourse to limits. By replacing infinitesimal ratios with finite q-differences, it constructs a parallel mathematical universe for non-continuous domains. The pioneering work of F. H. Jackson, who first systematized q-derivatives and integrals [1,2], demonstrated that this is not merely a discrete approximation but a genuine extension of classical calculus; as the parameter $q\to 1$, all classical results are recovered. The utility of this framework is vast, providing essential tools in fields as diverse as number theory and combinatorics through q-series and q-polynomials and mathematical physics [3,4,5]. Notably, its application extends to core scientific models, such as Fock’s [6] use of q-difference equations to analyze the symmetry of the hydrogen atom, underscoring its relevance beyond pure abstraction.

The natural progression of this generalization is post-quantum calculus, or $(p,q)$-calculus, which introduces a second independent parameter p. This bifactorial framework offers a richer and more flexible structure for "quantization," allowing for finer control and more nuanced generalizations of classical operators. It has become an instrumental language in the study of special functions, orthogonal polynomials, and operator theory. The definitions of $(p,q)$-derivative and $(p,q)$-integral in finite intervals and their fundamental properties were provided by Tunc and Gov [7,8]. Later, Soontharanon and Sitthiwirattham [9] defined $(p,q)$-difference operators with interval $[0,T]$ for $T>0$ and established basic properties of fractional $(p,q)$-calculus in this interval. In particular, they developed Riemann–Liouville and Caputo-type fractional $(p,q)$-integral operators. In this regard, Neang et al. [10] developed a new definition of the Riemann–Liouville fractional $(p,q)$ integral with respect to the q-shifting operator.

Numerical integration, the cornerstone of computational mathematics, provides indispensable tools for approximating definite integrals when exact analytic solutions are intractable. Classical quadrature rules, such as the midpoint, trapezoidal, Simpson, and Boole, supply efficient approximations accompanied by well-understood error estimates. These error bounds, often expressed as inequalities under smoothness or convexity assumptions, have been extensively studied and form a rich body of results known as quadrature inequalities. Notable among them are the Hermite–Hadamard inequality, Bullen–Simpson-type bounds, companion Ostrowski inequalities, and corrected Euler–Maclaurin formulas, each refining our understanding of how discrete sums relate to continuous integrals. In recent years, many quadrature inequalities have been constructed using various new strategies. Franjic and Pecaric [11] proved the corrected Euler–Maclaurin formulas and utilized the derived formulas to obtain inequalities for various classes of functions. Guessab and Schmeisser [12] proved companion Ostrowski-type integral inequalities under several regularity conditions by introducing the best constants in bounds of error and the remainder of the trapezoidal rule. Krukowski [13] introduced a natural successor to Simpson’s rule, which included Boole’s rule and provided novel error bounds for it. Meftah and Samoudi [14] established Bullen–Simpson-type inequalities for functions whose local fractional derivatives are s-convex functions and proved theorems when the first derivatives are bounded and Holderian. They also applied their results to numerical integration and special means. Sitthiwirattham et al. [15,16] determined error bounds related to Maclaurin’s formula using q-differentiable convex functions.

The integral inequalities related to Riemann–Liouville integrals have proven to be very valuable instruments in analysis in fractional calculus. These inequalities have enabled researchers to express stability conditions and constraints of fractional differential equations in a very effective manner and have also allowed researchers to shed ample light on the behavior of fractional equations in practical situations. It is also worth mentioning integral inequalities, which are obtained through the use of the Riemann–Liouville integral. Demir and Unes [17] developed corrected Euler–Maclaurin-type inequalities for differentiable convex functions, bounded functions, Lipschitzian functions, and bounded variation functions and demonstrated the precision of the developed results with numerical and graphical examples. Erden et al. [18] established fractional Ostrowski-type inequalities and obtained midpoint versions as a special case. Nasir et al. [19] studied a new type of Simpson-type inequalities using first-order locally differentiable convex functions and applied them to beta functions, q-digamma functions, and Bessel functions. Nasri et al. [20] derived the error estimates of fractional ${\displaystyle \frac{3}{8}}$ Simpson-type inequalities for functions whose absolute values of the first derivative are s-convex in the second sense and applied these findings to quadrature formulas, special means, and random variables. Samraiz et al. in [21] obtained fractional Bullen-type inequalities under the concept of h-convexity of twice-differentiable functions and applied these findings to special means and presented graphical examples. For further studies on integral inequalities involving the Riemann–Liouville integral, we recommend [22,23,24].

Post-quantum fractional integral inequalities represent a novel and rapidly evolving frontier in mathematical analysis, forging a critical connection between two profound areas of study: the geometric framework of quantum calculus and the nonlocal nature of fractional operators. This emerging discipline seeks to generalize classical analytic inequalities into a unified framework that incorporates both the discrete, parameter-dependent calculus of quantum operators and the continuous memory effects inherent to fractional integration and differentiation. Our main contributions to this paper include the following.

• Develop a new multiparameter identity that brings together $(p,q)$-derivative, $(p,q)$-Riemann–Liouville fractional integral, q-shifting operator, and q-Pochhammer symbol into one unified framework;
• Show that by simply choosing different parameter values, this single identity generates a wide family of known identities, such as the Boole-type, Bullen–Simpson-type, Maclaurin-type, corrected Euler–Maclaurin-type, ${\displaystyle \frac{3}{8}}$ Simpson-type, and companion Ostrowski-type;
• Successfully extend the classical work of B. Meftah and C. Menai to the broader world of post-quantum calculus, with their original results appearing as special cases;
• Create new integral inequalities using key mathematical tools such as modulus properties, s-convexity, log-convexity, power mean inequality, and Hölder’s inequality;
• Confirm all results with concrete numerical examples and graphs, visually proving that the inequalities hold for any choice of parameters;
• Showcase the application of the work through special means.

2. Preliminaries

Throughout the discussion $\mathbb{R}$ is considered to be the set of real numbers, and $0<q<p\le 1.$ Furthermore, in this section, it is assumed that $[{\epsilon }_1,{\epsilon }_3]\subset \mathbb{R}$ for ${\epsilon }_1<{\epsilon }_3.$ The $(p,q)$-number is defined as [10]

$${\left[m\right]}_{p,q}={\displaystyle \frac{p^m-q^m}{p-q}}=p^{m-1}{q}^0+p^{m-2}q+\cdots +p{q}^{m-2}+p^0{q}^{m-1},(m\in \mathbb{N}).$$

More generally [25],

$${\left[\upsilon \right]}_{p,q}={\displaystyle \frac{p^{\upsilon }-q^{\upsilon }}{p-q}},(\upsilon \in \mathbb{R}).$$

A q-shifting operator is defined as [10]

$${}_{{\epsilon }_1}{\Phi }_q\left(\upsilon \right)=q\upsilon +(1-q){\epsilon }_1,(\upsilon \in \mathbb{R}).$$

Further, we have

$${}_{{\epsilon }_1}{\Phi }_q^0\left(\upsilon \right)=\upsilon \mathrm{and}{}_{{\epsilon }_1}{\Phi }_q^m\left(\upsilon \right)={}_{{\epsilon }_1}{\Phi }_q^{m-1}\left({}_{{\epsilon }_1}{\Phi }_q\left(\upsilon \right)\right)={}_{{\epsilon }_1}{\Phi }_{q^m}\left(\upsilon \right),\left(m\in \mathbb{N}\right).$$

The q-Pochhammer symbol is defined as [26]

$${(\upsilon ;q)}_0=1,{(\upsilon ;q)}_k:=\prod _{\tau =0}^{k-1}\left(1-q^{\tau }\upsilon \right),(\upsilon \in \mathbb{R},k\in \mathbb{N}\cup \{\infty \}),$$

and the q-power function in the form of q-shifting operator is defined as

$${(c-\upsilon )}_{{\epsilon }_1}^{\left(0\right)}=1,{(c-\upsilon )}_{{\epsilon }_1}^{\left(k\right)}:=\prod _{\tau =0}^{k-1}\left(c-{}_{{\epsilon }_1}{\Phi }_q^{\tau }\left(\upsilon \right)\right),(\upsilon ,c\in \mathbb{R},k\in \mathbb{N}\cup \{\infty \}).$$

More generally, for $k=\varphi \in \mathbb{R}$

$${(c-\upsilon )}_{{\epsilon }_1}^{\left(\varphi \right)}=c^{\varphi }\prod _{\tau =0}^{\infty }{\displaystyle \frac{1-\left(\frac{\upsilon }{c}\right)q^{\tau }}{1-\left(\frac{\upsilon }{c}\right)q^{\varphi +\tau }}},(c\ne 0).$$

Definition 1  

([27]). Let $\aleph :[{\epsilon }_1,{\epsilon }_3]\to \mathbb{R}$ be a continuous function. Then, the $(p,q){}_{{\epsilon }_1}$-derivative of ℵ at σ is given by

$$\begin{array}{c}\hfill {}_{{\epsilon }_1}{D}_{p,q}\aleph \left(\sigma \right)=\left\{\begin{array}{cc}{\displaystyle \frac{\aleph (p\sigma +(1-p){\epsilon }_1)-\aleph (q\sigma +(1-q){\epsilon }_1)}{(p-q)(\sigma -{\epsilon }_1)}},\hfill & \sigma \ne {\epsilon }_1;\hfill \\ \underset{\sigma \to {\epsilon }_1}{lim}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left(\sigma \right),\hfill & \sigma ={\epsilon }_1.\hfill \end{array}\right.\end{array}$$

The function ℵ is called $(p,q){}_{{\epsilon }_1}$-differentiable on $[{\epsilon }_1,{\epsilon }_3]$ if ${}_{{\epsilon }_1}{D}_{p,q}\aleph \left(\sigma \right)$ exists for all $\sigma \in [{\epsilon }_1,{\epsilon }_1+{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{p}}].$

Definition 2 

([27]). Let $\aleph :[{\epsilon }_1,{\epsilon }_3]\to \mathbb{R}$ be a continuous function. Then, the $(p,q){}_{{\epsilon }_1}$-integral of ℵ at σ is given by

$$\begin{array}{c}\hfill {\int }_{{\epsilon }_1}^{{\epsilon }_3}\aleph \left(\sigma \right){}_{{\epsilon }_1}{d}_{p,q}\sigma =(p-q)({\epsilon }_3-{\epsilon }_1)\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}\aleph \left({\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}{\epsilon }_3+\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}\right){\epsilon }_1\right).\end{array}$$

The function ℵ is called $(p,q){}_{{\epsilon }_1}$-integrable on $[{\epsilon }_1,{\epsilon }_3]$ if ${\int }_{{\epsilon }_1}^{{\epsilon }_3}\aleph \left(\sigma \right){}_{{\epsilon }_1}{d}_{p,q}\sigma$ exists for all $\sigma \in [{\epsilon }_1,{\epsilon }_1+p({\epsilon }_3-{\epsilon }_1\left)\right].$

The $(p,q)$-power function is defined by [10]

$${}_{{\epsilon }_1}{(\upsilon -c)}_{p,q}^{\left(0\right)}:=1,{}_{{\epsilon }_1}{(\upsilon -c)}_{p,q}^{\left(k\right)}:=\prod _{\tau =0}^{k-1}\left({}_{{\epsilon }_1}{\Phi }_p^{\tau }\left(\upsilon \right)-{}_{{\epsilon }_1}{\Phi }_q^{\tau }\left(c\right)\right),\left(\upsilon ,c\in \mathbb{R},k\in {\mathbb{N}}_0\right).$$

It can be observed that

$${}_{{\epsilon }_1}{(\upsilon -c)}_{p,q}^{\left(k\right)}={(\upsilon -{\epsilon }_1)}^k\prod _{\tau =0}^{k-1}{p}^{\tau }\left(1-\left({\displaystyle \frac{c-{\epsilon }_1}{\upsilon -{\epsilon }_1}}\right){\left({\displaystyle \frac{q}{p}}\right)}^{\tau }\right).$$

More generally, for $k=\varphi \in \mathbb{R},$

$$\begin{array}{c}\hfill {}_{{\epsilon }_1}{(\upsilon -c)}_{p,q}^{\left(\varphi \right)}={(\upsilon -{\epsilon }_1)}^{\varphi }\prod _{\tau =0}^{\infty }{\displaystyle \frac{p^{\tau }}{p^{\varphi +\tau }}}{\displaystyle \frac{1-\left(\frac{c-{\epsilon }_1}{\upsilon -{\epsilon }_1}\right){\left(\frac{q}{p}\right)}^{\tau }}{1-\left(\frac{c-{\epsilon }_1}{\upsilon -{\epsilon }_1}\right){\left(\frac{q}{p}\right)}^{\varphi +\tau }}}\\ \hfill ={(\upsilon -{\epsilon }_1)}^{\varphi }{p}^{\left({\displaystyle \genfrac{}{}{0pt}{}{\varphi }{2}}\right)}\prod _{\tau =0}^{\infty }{\displaystyle \frac{1-\left(\frac{c-{\epsilon }_1}{\upsilon -{\epsilon }_1}\right){\left(\frac{q}{p}\right)}^{\tau }}{1-\left(\frac{c-{\epsilon }_1}{\upsilon -{\epsilon }_1}\right){\left(\frac{q}{p}\right)}^{\varphi +\tau }}},\end{array}$$

(1)

particularly for $\varphi >0,$

$$\begin{array}{c}\hfill {}_{{\epsilon }_1}{\left(\upsilon -{}_{{\epsilon }_1}{\Phi }_{q/p}^k\left(\upsilon \right)\right)}_{p,q}^{\left(\varphi \right)}={(\upsilon -{\epsilon }_1)}^{\varphi }\prod _{\tau =0}^{\infty }{\displaystyle \frac{p^{\tau }}{p^{\varphi +\tau }}}{\displaystyle \frac{1-{\left(\frac{q}{p}\right)}^{k+\tau }}{1-{\left(\frac{q}{p}\right)}^{k+\varphi +\tau }}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={(\upsilon -{\epsilon }_1)}^{\varphi }{\left(1-{\left({\displaystyle \frac{q}{p}}\right)}^k\right)}_{p,q}^{\left(\varphi \right)}.\hfill \end{array}$$

(2)

The $(p,q)$-gamma function is defined by [10]

$${\Gamma }_{p,q}\left(\varkappa \right)={\displaystyle \frac{{(p-q)}_{p,q}^{(\varkappa -1)}}{{(p-q)}^{\varkappa -1}}},\left(\varkappa \in \mathbb{R}\setminus \{0,-1,-2,-3,\cdots \}\right).$$

(3)

P. N. Sadjang [28] provided the equivalent definition of (3) as follows:

$${\Gamma }_{p,q}\left(\varkappa \right)=p^{{\displaystyle \frac{\varkappa (\varkappa -1)}{2}}}{\int }_0^{\infty }{\sigma }^{\varkappa -1}{E}_{p,q}^{-q\sigma }{d}_{p,q}\sigma ,$$

provided that

$$E_{p,q}^{-q\sigma }=\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\left(\genfrac{}{}{0pt}{}{\tau }{2}\right)}{(-q\sigma )}^{\tau }}{{\left[\tau \right]}_{p,q}}}.$$

Obviously, ${\Gamma }_{p,q}(\varkappa +1)={\left[\varkappa \right]}_{p,q}{\Gamma }_{p,q}\left(\varkappa \right).$ The $(p,q)$-beta function is defined by [10]

$$B_{p,q}(\varrho ,\varkappa )={\int }_0^1{\sigma }^{\varrho -1}{}_0{\left(1-{}_0{\Phi }_q\left(\sigma \right)\right)}_{p,q}^{(\varkappa -1)}{}_0{d}_{p,q}\sigma ,(\varrho ,\varkappa >0),$$

(4)

and another way to write (4) is as follows:

$$B_{p,q}(\varrho ,\varkappa )=p^{{\displaystyle \frac{(\varkappa -1)(2\varrho +\varkappa -2)}{2}}}{\displaystyle \frac{{\Gamma }_{p,q}\left(\varrho \right){\Gamma }_{p,q}\left(\varkappa \right)}{{\Gamma }_{p,q}(\varrho +\varkappa )}}.$$

Definition 3  

([26]). Let ℵ be defined on $[{\epsilon }_1,{\epsilon }_3],$ and let $\varphi >0.$ The Riemann–Liouville fractional $q{}_{{\epsilon }_1}$-integral is defined by

$$\begin{array}{ccc}\hfill {}_{{\epsilon }_1}{J}_q^{\varphi }\aleph \left(\psi \right)& :=& {\displaystyle \frac{1}{{\Gamma }_q\left(\varphi \right)}}{\int }_{{\epsilon }_1}^{\psi }{}_{{\epsilon }_1}{\left(\psi -{}_{{\epsilon }_1}{\Phi }_q\left(\sigma \right)\right)}_q^{(\varphi -1)}\aleph \left(\sigma \right){}_{{\epsilon }_1}{d}_q\sigma \hfill \\ & =& {\displaystyle \frac{(1-q)(\psi -{\epsilon }_1)}{{\Gamma }_q\left(\varphi \right)}}\sum _{\tau =0}^{\infty }{q}^{\tau }{}_{{\epsilon }_1}{\left(\psi -{}_{{\epsilon }_1}{\Phi }_q^{\tau +1}\left(\psi \right)\right)}_q^{(\varphi -1)}\aleph \left({}_{{\epsilon }_1}{\Phi }_q^{\tau }\left(\psi \right)\right).\hfill \end{array}$$

Definition 4  

([10]). Let ℵ be defined on $[{\epsilon }_1,{\epsilon }_3],$ and let $\varphi >0.$ The Riemann–Liouville fractional $(p,q){}_{{\epsilon }_1}$-integral is defined by

$$\begin{array}{ccc}\hfill {}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \left(\psi \right)& :=& {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}{\Gamma }_{p,q}\left(\varphi \right)}}{\int }_{{\epsilon }_1}^{\psi }{}_{{\epsilon }_1}{\left(\psi -{}_{{\epsilon }_1}{\Phi }_q\left(\sigma \right)\right)}_{p,q}^{(\varphi -1)}\aleph \left({}_{{\epsilon }_1}{\Phi }_p^{1-\varphi }\left(\sigma \right)\right){}_{{\epsilon }_1}{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{(p-q)\left(\psi -{\epsilon }_1\right)}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}{\Gamma }_{p,q}\left(\varphi \right)}}\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}{}_{{\epsilon }_1}{\left(\psi -{}_{{\epsilon }_1}{\Phi }_{q/p}^{\tau +1}\left(\psi \right)\right)}_{p,q}^{(\varphi -1)}\hfill \\ & & \times \aleph \left({\displaystyle \frac{q^{\tau }}{p^{\varphi +\tau }}}\psi +\left(1-{\displaystyle \frac{q^{\tau }}{p^{\varphi +\tau }}}\right){\epsilon }_1\right),\hfill \end{array}$$

provided that $\psi \in [{\epsilon }_1,{\epsilon }_1+p^{\varphi }({\epsilon }_3-{\epsilon }_1)].$

Remark 1.  

When $p=1$, the Definition 4 coincides with the Definition 3.

3. Main Results

Lemma 1.  

Let $\aleph :[{\epsilon }_1,{\epsilon }_3]\subset \mathbb{R}\to \mathbb{R}$ be ${(p,q)}^{{\epsilon }_1},$${(p,q)}^{{\epsilon }_2},$${(p,q)}^{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}$ and ${(p,q)}^{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}$ differentiable function on $I_1,$$I_2,$$I_3$ and $I_4$, respectively, such that ${}_{{\epsilon }_1}{D}_{p,q}\aleph ,$${}_{{\epsilon }_2}{D}_{p,q}\aleph ,$${}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph$ and ${}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph$ are continuous and integrable, respectively, on $J_1,$$J_2,$$J_3$ and $J_4$ and $\varphi >0,$${\epsilon }_2\in \left({\epsilon }_1,{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right),$ and $\alpha ,\beta \in [0,1],$ then

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1\left({\displaystyle \frac{1+\beta -\alpha }{1+\beta }}-{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right){}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left({\displaystyle \frac{\beta ({\epsilon }_3-{\epsilon }_1)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}-{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right)\hfill \\ & & \times {}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left({\displaystyle \frac{{\epsilon }_3+{\epsilon }_1+2(\beta {\epsilon }_1-\beta {\epsilon }_2-{\epsilon }_2)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}-{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right)\hfill \\ & & \times {}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1\left({\displaystyle \frac{\alpha }{1+\beta }}-{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right)\hfill \\ & & \times {}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & & ={\displaystyle \frac{\alpha ({\epsilon }_2-{\epsilon }_1)}{({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\beta }{1+\beta }}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1-2\alpha ({\epsilon }_2-{\epsilon }_1)}{2({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\hfill \\ & & -{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^{1-\varphi }{\Gamma }_{p,q}(\varphi +1)}{({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}\hfill \\ & & -{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1-\varphi }{\Gamma }_{p,q}(\varphi +1)}{2^{1-\varphi }({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\},\hfill \end{array}$$

(5)

where

$$I_1:=\left({\epsilon }_1,{\epsilon }_1+{\displaystyle \frac{{\epsilon }_2-{\epsilon }_1}{p}}\right),I_2:=\left({\epsilon }_2,{\epsilon }_2+{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2p}}\right),$$

$$I_3:=\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}},{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2p}}\right),I_4:=\left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2,{\epsilon }_1+{\epsilon }_3-{\epsilon }_2+{\displaystyle \frac{{\epsilon }_2-{\epsilon }_1}{p}}\right),$$

$$J_1:=\left({\epsilon }_1,{\epsilon }_1+p({\epsilon }_2-{\epsilon }_1)\right),J_2:=\left({\epsilon }_2,{\epsilon }_2+p{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2}}\right),$$

$$J_3:=\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}},{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+p{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2}}\right),J_4:=\left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2,{\epsilon }_1+{\epsilon }_3-{\epsilon }_2+p({\epsilon }_2-{\epsilon }_1)\right).$$

Proof. 

According to Definition 1,

$$\begin{array}{c}\hfill {}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)={\displaystyle \frac{\aleph \left((1-q\sigma ){\epsilon }_1+\sigma q{\epsilon }_2\right)-\aleph \left((1-p\sigma ){\epsilon }_1+\sigma p{\epsilon }_2\right)}{\sigma (p-q)({\epsilon }_1-{\epsilon }_2)}}.\end{array}$$

(6)

Using (6) and keeping ${}_{{\epsilon }_1}{D}_{p,q}\aleph ,$ continuous and integrable in $J_1=\left({\epsilon }_1,{\epsilon }_1+p({\epsilon }_2-{\epsilon }_1)\right)$ in Definition 2, produce the following:

$$\begin{array}{ccc}\hfill {\mathfrak{J}}_0& :=& {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{\int }_0^1{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{1}{(p-q)({\epsilon }_1-{\epsilon }_2)p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{\int }_0^1{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}\left\{\aleph \left((1-q\sigma ){\epsilon }_1+q\sigma {\epsilon }_2\right)\right.\hfill \\ & & \left.-\aleph \left((1-p\sigma ){\epsilon }_1+p\sigma {\epsilon }_2\right)\right\}{\displaystyle \frac{{}_0{d}_{p,q}\sigma }{\sigma }}\hfill \\ & =& {\displaystyle \frac{1}{(p-q)({\epsilon }_1-{\epsilon }_2)p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\left[{\int }_0^1{\left(1-q\sigma \right)}_{p,q}^{\left(\varphi \right)}\aleph \left((1-q\sigma ){\epsilon }_1+q\sigma {\epsilon }_2\right){\displaystyle \frac{{}_0{d}_{p,q}\sigma }{\sigma }}\right.\hfill \\ & & \left.-{\int }_0^1{\left(1-q\sigma \right)}_{p,q}^{\left(\varphi \right)}\aleph \left((1-p\sigma ){\epsilon }_1+p\sigma {\epsilon }_2\right){\displaystyle \frac{{}_0{d}_{p,q}\sigma }{\sigma }}\right]\hfill \\ & =& {\displaystyle \frac{1}{({\epsilon }_1-{\epsilon }_2)p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}{\left(1-q{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}\right)}_{p,q}^{\left(\varphi \right)}\left[{\displaystyle \frac{\aleph \left(\left(1-q\frac{q^{\tau }}{p^{\tau +1}}\right){\epsilon }_1+q\frac{q^{\tau }}{p^{\tau +1}}{\epsilon }_2\right)}{\frac{q^{\tau }}{p^{\tau +1}}}}\right.\hfill \\ & & \left.-{\displaystyle \frac{\aleph \left(\left(1-p\frac{q^{\tau }}{p^{\tau +1}}\right){\epsilon }_1+p\frac{q^{\tau }}{p^{\tau +1}}{\epsilon }_2\right)}{\frac{q^{\tau }}{p^{\tau +1}}}}\right]\hfill \\ & =& {\displaystyle \frac{1}{({\epsilon }_1-{\epsilon }_2)p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\sum _{\tau =0}^{\infty }{\left(1-{\left({\displaystyle \frac{q}{p}}\right)}^{\tau +1}\right)}_{p,q}^{\left(\varphi \right)}\left[\aleph \left(\left(1-{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}{\epsilon }_2\right)\right.\hfill \\ & & \left.-\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right].\hfill \end{array}$$

Utilizing (1) and (2) along with the q-Pochhammer symbol definition, produce the following:

$$\begin{array}{ccc}\hfill {\mathfrak{J}}_0& =& {\displaystyle \frac{1}{({\epsilon }_1-{\epsilon }_2)p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\sum _{\tau =0}^{\infty }\left(p^{\left({\displaystyle \genfrac{}{}{0pt}{}{\varphi }{2}}\right)}\prod _{k=0}^{\infty }{\displaystyle \frac{1-{\left(\frac{q}{p}\right)}^{\tau +k+1}}{1-{\left(\frac{q}{p}\right)}^{\tau +\varphi +k+1}}}\right)\left[\aleph \left(\left(1-{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}{\epsilon }_2\right)\right.\hfill \\ & & \left.-\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[\sum _{\tau =0}^{\infty }{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +1+\varphi }}{p^{\tau +1+\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}{\epsilon }_2\right)\right.\hfill \\ & & \left.-\sum _{\tau =0}^{\infty }{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +1+\varphi }}{p^{\tau +1+\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[-\sum _{\tau =0}^{\infty }\left(1-{\displaystyle \frac{q^{\tau +\varphi }}{p^{\tau +\varphi }}}\right){\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right.\hfill \\ & & \left.+\sum _{\tau =0}^{\infty }\left(1-{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}\right){\displaystyle \frac{{\left(\frac{q^{\tau +2}}{p^{\tau +2}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +1+\varphi }}{p^{\tau +1+\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}{\epsilon }_2\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[-\sum _{\tau =0}^{\infty }{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right.\hfill \\ \hfill & & \left.+\sum _{\tau =0}^{\infty }{\displaystyle \frac{{\left(\frac{q^{\tau +2}}{p^{\tau +2}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +1+\varphi }}{p^{\tau +1+\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}{\epsilon }_2\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau +\varphi }}{p^{\tau +\varphi }}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right.\hfill \\ & & \left.-\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}{\displaystyle \frac{{\left(\frac{q^{\tau +2}}{p^{\tau +2}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +1+\varphi }}{p^{\tau +1+\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau +1}}{p^{\tau +1}}}{\epsilon }_2\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[\aleph \left({\epsilon }_1\right)-{\displaystyle \frac{{\left(\frac{q}{p};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\varphi }}{p^{\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left({\epsilon }_2\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau +\varphi }}{p^{\tau +\varphi }}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right.\hfill \\ & & \left.-\sum _{\tau =1}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[\aleph \left({\epsilon }_1\right)-{\displaystyle \frac{{\left(\frac{q}{p};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\varphi }}{p^{\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left({\epsilon }_2\right)\right]\hfill \\ & & +{\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left[\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau +\varphi }}{p^{\tau +\varphi }}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right.\hfill \\ & & \left.-\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)+{\displaystyle \frac{{\left(\frac{q}{p};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\varphi }}{p^{\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left({\epsilon }_2\right)\right]\hfill \\ & =& {\displaystyle \frac{\aleph \left({\epsilon }_1\right)}{{\epsilon }_1-{\epsilon }_2}}-{\displaystyle \frac{1}{{\epsilon }_1-{\epsilon }_2}}\left(1-{\displaystyle \frac{q^{\varphi }}{p^{\varphi }}}\right)\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\hfill \\ & =& {\displaystyle \frac{\aleph \left({\epsilon }_1\right)}{{\epsilon }_1-{\epsilon }_2}}+{\displaystyle \frac{{\left[\varphi \right]}_{p,q}(p-q)}{({\epsilon }_2-{\epsilon }_1)p^{\varphi }}}\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\hfill \\ & =& {\displaystyle \frac{\aleph \left({\epsilon }_1\right)}{{\epsilon }_1-{\epsilon }_2}}+{\displaystyle \frac{p^{1+\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}{\Gamma }_{p,q}(\varphi +1)}{p^{{\varphi }^2+\varphi +\frac{\varphi -1}{2}}{({\epsilon }_2-{\epsilon }_1)}^{\varphi +1}}}\times {\displaystyle \frac{(p-q)\left[p^{\varphi }({\epsilon }_2-{\epsilon }_1)\right]}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}{\Gamma }_{p,q}\left(\varphi \right)}}\hfill \\ & & \times \sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}{p}^{{\displaystyle \frac{\varphi -1}{2}}}{\left(p^{\varphi }({\epsilon }_2-{\epsilon }_1)\right)}^{\varphi -1}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\hfill \\ & =& {\displaystyle \frac{\aleph \left({\epsilon }_1\right)}{{\epsilon }_1-{\epsilon }_2}}+{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left[{\displaystyle \frac{(p-q)p^{\varphi }({\epsilon }_2-{\epsilon }_1)}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}{\Gamma }_{p,q}\left(\varphi \right)}}\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}{\displaystyle \frac{{\left(\frac{q^{\tau +1}}{p^{\tau +1}};\frac{q}{p}\right)}_{\infty }}{{\left(\frac{q^{\tau +\varphi }}{p^{\tau +\varphi }};\frac{q}{p}\right)}_{\infty }}}\right.\hfill \\ & & \left.\times p^{{\displaystyle \frac{\varphi -1}{2}}}{\left(p^{\varphi }({\epsilon }_2-{\epsilon }_1)\right)}^{\varphi -1}\aleph \left(\left(1-{\displaystyle \frac{q^{\tau }}{p^{\tau }}}\right){\epsilon }_1+{\displaystyle \frac{q^{\tau }}{p^{\tau }}}{\epsilon }_2\right)\right]\hfill \\ \hfill & =& {\displaystyle \frac{\aleph \left({\epsilon }_1\right)}{{\epsilon }_1-{\epsilon }_2}}+{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\times {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}{\Gamma }_{p,q}\left(\varphi \right)}}{\int }_{{\epsilon }_1}^{{}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)}{}_{{\epsilon }_1}{\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)-{}_{{\epsilon }_1}{\Phi }_q\left(s\right)\right)}_{p,q}^{(\varphi -1)}\hfill \\ & & \times \aleph \left({\displaystyle \frac{s}{p^{\varphi -1}}}+\left(1-{\displaystyle \frac{1}{p^{\varphi -1}}}\right){\epsilon }_1\right){}_{{\epsilon }_1}{d}_{p,q}s\hfill \\ & =& {\displaystyle \frac{\aleph \left({\epsilon }_1\right)}{{\epsilon }_1-{\epsilon }_2}}+{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right).\hfill \end{array}$$

(7)

Similarly,

$$\begin{array}{ccc}\hfill {\mathfrak{J}}_1& :=& {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{\int }_0^1{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{2\aleph \left({\epsilon }_2\right)}{2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3}}+{\displaystyle \frac{2^{1+\varphi }{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathfrak{J}}_2& :=& {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{\int }_0^1{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{2\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)}{2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3}}+{\displaystyle \frac{2^{1+\varphi }{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right).\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathfrak{J}}_3& :=& {\displaystyle \frac{1}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{\int }_0^1{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)}{{\epsilon }_1-{\epsilon }_2}}+{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right).\hfill \end{array}$$

Now,

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_0& :=& {\displaystyle \frac{1+\beta -\alpha }{1+\beta }}{\int }_0^1{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{1+\beta -\alpha }{(1+\beta )(p-q)({\epsilon }_1-{\epsilon }_2)}}{\int }_0^1\left\{\aleph \left((1-q\sigma ){\epsilon }_1+q\sigma {\epsilon }_2\right)\right.\left.-\aleph \left((1-p\sigma ){\epsilon }_1+p\sigma {\epsilon }_2\right)\right\}{\displaystyle \frac{{}_0{d}_{p,q}\sigma }{\sigma }}\hfill \\ & =& {\displaystyle \frac{1+\beta -\alpha }{(1+\beta )({\epsilon }_1-{\epsilon }_2)}}\sum _{\tau =0}^{\infty }{\displaystyle \frac{q^{\tau }}{p^{\tau +1}}}\left[{\displaystyle \frac{\aleph \left(\left(1-{}_0{\Phi }_{q/p}^{\tau +1}\left(1\right)\right){\epsilon }_1+{}_0{\Phi }_{q/p}^{\tau +1}\left(1\right){\epsilon }_2\right)}{\frac{q^{\tau }}{p^{\tau +1}}}}\right.\hfill \\ & & \left.-{\displaystyle \frac{\aleph \left(\left(1-{}_0{\Phi }_{q/p}^{\tau }\left(1\right)\right){\epsilon }_1+{}_0{\Phi }_{q/p}^{\tau }\left(1\right){\epsilon }_2\right)}{\frac{q^{\tau }}{p^{\tau +1}}}}\right]\hfill \\ & =& {\displaystyle \frac{1+\beta -\alpha }{(1+\beta )({\epsilon }_1-{\epsilon }_2)}}\left\{\aleph \left({\epsilon }_1\right)-\aleph \left({\epsilon }_2\right)\right\}.\hfill \end{array}$$

(8)

Similarly,

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_1& :=& {\displaystyle \frac{\beta ({\epsilon }_3-{\epsilon }_1)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}{\int }_0^1{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{2\beta ({\epsilon }_3-{\epsilon }_1)}{(1+\beta ){({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}}\left\{\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)-\aleph \left({\epsilon }_2\right)\right\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_2& :=& {\displaystyle \frac{{\epsilon }_3+{\epsilon }_1+2\beta {\epsilon }_1-2\beta {\epsilon }_2-2{\epsilon }_2}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}{\int }_0^1{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{2({\epsilon }_3+{\epsilon }_1+2\beta {\epsilon }_1-2\beta {\epsilon }_2-2{\epsilon }_2)}{(1+\beta ){({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}}\left\{\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)-\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right\}.\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_3& :=& {\displaystyle \frac{\alpha }{1+\beta }}{\int }_0^1{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & =& {\displaystyle \frac{\alpha }{(1+\beta )({\epsilon }_1-{\epsilon }_2)}}\left\{\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)-\aleph \left({\epsilon }_3\right)\right\}.\hfill \end{array}$$

Subtracting (7) from (8)

$$\begin{array}{ccc}\hfill {\mathfrak{L}}_0-{\mathfrak{J}}_0& =& {\displaystyle \frac{\alpha \aleph \left({\epsilon }_1\right)+(1+\beta -\alpha )\aleph \left({\epsilon }_2\right)}{(1+\beta )({\epsilon }_2-{\epsilon }_1)}}-{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right).\hfill \end{array}$$

Equivalently,

$$\begin{array}{c}{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2)}{{\epsilon }_3-{\epsilon }_1}}({\mathfrak{L}}_0-{\mathfrak{J}}_0)={\displaystyle \frac{({\epsilon }_2-{\epsilon }_1)\left\{\alpha \aleph \left({\epsilon }_1\right)+(1+\beta -\alpha )\aleph \left({\epsilon }_2\right)\right\}}{(1+\beta )({\epsilon }_3-{\epsilon }_1)}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^{1-\varphi }{\Gamma }_{p,q}(\varphi +1)}{({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right).\hfill \end{array}$$

(9)

Similarly,

$$\begin{array}{c}{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}({\mathfrak{L}}_1-{\mathfrak{J}}_1)\hfill \\ \hspace{1em}\hspace{1em}={\displaystyle \frac{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2+2\beta {\epsilon }_1-2\beta {\epsilon }_2)\aleph \left({\epsilon }_2\right)+\beta ({\epsilon }_3-{\epsilon }_1)\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)}{2({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1-\varphi }{\Gamma }_{p,q}(\varphi +1)}{2^{1-\varphi }({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right).\end{array}$$

(10)

$$\begin{array}{c}{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}({\mathfrak{L}}_2-{\mathfrak{J}}_2)\hfill \\ \hspace{1em}\hspace{1em}={\displaystyle \frac{\beta ({\epsilon }_3-{\epsilon }_1)\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)+({\epsilon }_1+{\epsilon }_3+2\beta {\epsilon }_1-2\beta {\epsilon }_2-2{\epsilon }_2)\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)}{2({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1-\varphi }{\Gamma }_{p,q}(\varphi +1)}{2^{1-\varphi }({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right).\end{array}$$

(11)

$$\begin{array}{c}{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}({\mathfrak{L}}_3-{\mathfrak{J}}_3)={\displaystyle \frac{({\epsilon }_2-{\epsilon }_1)\left\{(1+\beta -\alpha )\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\alpha \aleph \left({\epsilon }_3\right)\right\}}{(1+\beta )({\epsilon }_3-{\epsilon }_1)}}\hfill \\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^{1-\varphi }{\Gamma }_{p,q}(\varphi +1)}{({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right).\end{array}$$

(12)

Addition of (9)–(12) yields the identity (5). □

Remark 2. 

•      For $p=1,$ identity (5) reduces to Riemann–Liouville fractional quantum identity:

$$\begin{array}{ccc}& & {\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1\left({\displaystyle \frac{1+\beta -\alpha }{1+\beta }}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_q^{\left(\varphi \right)}\right){}_{{\epsilon }_1}{D}_q\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right){}_0{d}_q\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left({\displaystyle \frac{\beta ({\epsilon }_3-{\epsilon }_1)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_q^{\left(\varphi \right)}\right)\hfill \\ & & \times {}_{{\epsilon }_2}{D}_q\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_q\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left({\displaystyle \frac{{\epsilon }_3+{\epsilon }_1+2(\beta {\epsilon }_1-\beta {\epsilon }_2-{\epsilon }_2)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_q^{\left(\varphi \right)}\right)\hfill \\ & & \times {}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right){}_0{d}_q\sigma +{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}\hfill \\ & & \times {\int }_0^1\left({\displaystyle \frac{\alpha }{1+\beta }}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_q^{\left(\varphi \right)}\right){}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right){}_0{d}_q\sigma \hfill \\ & & ={\displaystyle \frac{\alpha ({\epsilon }_2-{\epsilon }_1)}{({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\beta }{1+\beta }}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1-2\alpha ({\epsilon }_2-{\epsilon }_1)}{2({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\hfill \\ & & -{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^{1-\varphi }{\Gamma }_q(\varphi +1)}{({\epsilon }_3-{\epsilon }_1)}}\left\{\left({}_{{\epsilon }_1}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_2\right)+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_3\right)\right\}\hfill \\ & & -{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1-\varphi }{\Gamma }_q(\varphi +1)}{2^{1-\varphi }({\epsilon }_3-{\epsilon }_1)}}\left\{\left({}_{{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{J}_q^{\varphi }\aleph \right)({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}.\hfill \end{array}$$

Moreover, for $\varphi =1,$ it reduces to

$$\begin{array}{ccc}& & {\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1\left(q\sigma -{\displaystyle \frac{\alpha }{1+\beta }}\right){}_{{\epsilon }_1}{D}_q\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right){}_0{d}_q\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left(q\sigma -{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3+2(\beta {\epsilon }_1-\beta {\epsilon }_2-{\epsilon }_2)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}\right)\hfill \\ & & \times {}_{{\epsilon }_2}{D}_q\aleph \left((1-\sigma ){\epsilon }_2+t{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_q\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left(q\sigma -{\displaystyle \frac{\beta ({\epsilon }_3-{\epsilon }_1)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}\right)\hfill \\ & & \times {}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right){}_0{d}_q\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1\left(q\sigma -1+{\displaystyle \frac{\alpha }{1+\beta }}\right){}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right){}_0{d}_q\sigma \hfill \\ & & ={\displaystyle \frac{\alpha ({\epsilon }_2-{\epsilon }_1)}{({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\beta }{1+\beta }}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1-2\alpha ({\epsilon }_2-{\epsilon }_1)}{2({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}-{\displaystyle \frac{1}{{\epsilon }_3-{\epsilon }_1}}\left\{{\int }_{{\epsilon }_1}^{{\epsilon }_2}\aleph \left(s\right){}_{{\epsilon }_1}{d}_{q}s\right.\hfill \\ & & +\left.{\int }_{{\epsilon }_2}^{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}\aleph \left(s\right){}_{{\epsilon }_2}{d}_{q}s+{\int }_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}^{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}\aleph \left(s\right){}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{d}_{q}s+{\int }_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}^{{\epsilon }_3}\aleph \left(s\right){}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{d}_{q}s\right\},\hfill \end{array}$$

hence for $q\to 1,$ it reduces to Lemma 2.1 [29]:

$$\begin{array}{ccc}& & {\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1\left(\sigma -{\displaystyle \frac{\alpha }{1+\beta }}\right){\aleph }^{\prime }\left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)d\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left(\sigma -{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3+2(\beta {\epsilon }_1-\beta {\epsilon }_2-{\epsilon }_2)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}\right)\hfill \\ & & \times {\aleph }^{\prime }\left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)d\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1\left(\sigma -{\displaystyle \frac{\beta ({\epsilon }_3-{\epsilon }_1)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}}\right)\hfill \\ & & \times {\aleph }^{\prime }\left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)d\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1\left(\sigma -1+{\displaystyle \frac{\alpha }{1+\beta }}\right){\aleph }^{\prime }\left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)d\sigma \hfill \\ & & ={\displaystyle \frac{\alpha ({\epsilon }_2-{\epsilon }_1)}{({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\}+{\displaystyle \frac{\beta }{1+\beta }}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1-2\alpha ({\epsilon }_2-{\epsilon }_1)}{2({\epsilon }_3-{\epsilon }_1)(1+\beta )}}\left\{\aleph \left({\epsilon }_2\right)+\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right\}-{\displaystyle \frac{1}{{\epsilon }_3-{\epsilon }_1}}{\int }_{{\epsilon }_1}^{{\epsilon }_3}\aleph \left(s\right)ds.\hfill \end{array}$$

•      For $\alpha ={\displaystyle \frac{14}{39}},$$\beta ={\displaystyle \frac{2}{13}}$ and ${\epsilon }_2={\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}},$ identity (5) reduces to a post-quantum Boole-type identity:

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{16}}\left[{\int }_0^1{\displaystyle \frac{31p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-45{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{45p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}\right){}_0{d}_{p,q}\sigma \right.\hfill \\ & & +{\int }_0^1{\displaystyle \frac{4p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-15{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{15p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\int }_0^1{\displaystyle \frac{11p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-15{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{15p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma {\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & \left.+{\int }_0^1{\displaystyle \frac{14p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-45{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{45p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \right]\hfill \\ & & ={\displaystyle \frac{7\aleph \left({\epsilon }_1\right)+32\aleph \left(\frac{3{\epsilon }_1+{\epsilon }_3}{4}\right)+12\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)+32\aleph \left(\frac{{\epsilon }_1+3{\epsilon }_3}{4}\right)+7\aleph \left({\epsilon }_3\right)}{90}}\hfill \\ & & -{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{4^{1-\varphi }{({\epsilon }_3-{\epsilon }_1)}^{\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)\right.\hfill \\ & & +\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}\right)\right)+\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}\right)\right)\hfill \\ & & \left.+\left({}_{{\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}.\hfill \end{array}$$

•      For $\alpha ={\displaystyle \frac{2}{5}},$$\beta ={\displaystyle \frac{1}{5}}$ and ${\epsilon }_2={\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}},$ identity (5) reduces to a post-quantum Bullen–Simpson-type identity:

$$\begin{array}{ccc}& & {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{16}}\left[{\int }_0^1{\displaystyle \frac{2p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-3{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{3p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}\right){}_0{d}_{p,q}\sigma \right.\hfill \\ & & +{\int }_0^1{\displaystyle \frac{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-3{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{3p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\int }_0^1{\displaystyle \frac{2p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-3{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{3p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma {\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & \left.+{\int }_0^1{\displaystyle \frac{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-3{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{3p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}+t{\epsilon }_3\right){}_0{d}_{p,q}\sigma \right]\hfill \\ & & ={\displaystyle \frac{\aleph \left({\epsilon }_1\right)+4\aleph \left(\frac{3{\epsilon }_1+{\epsilon }_3}{4}\right)+2\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)+4\aleph \left(\frac{{\epsilon }_1+3{\epsilon }_3}{4}\right)+\aleph \left({\epsilon }_3\right)}{12}}\hfill \\ & & -{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{4^{1-\varphi }{({\epsilon }_3-{\epsilon }_1)}^{\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)\right.\hfill \\ & & +\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}\right)\right)+\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}\right)\right)\hfill \\ & & \left.+\left({}_{{\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}.\hfill \end{array}$$

•      For $\alpha =0,$$\beta ={\displaystyle \frac{1}{3}}$ and ${\epsilon }_2={\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}},$ identity (5) reduces to a post-quantum Maclaurin-type identity:

$$\begin{array}{ccc}& & {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36}}{\int }_0^1{\displaystyle \frac{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\int }_0^1{\displaystyle \frac{3p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-8{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{8p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\hfill \\ & & \times {}_{{\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\int }_0^1{\displaystyle \frac{5p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-8{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{8p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\hfill \\ & & \times {}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma {\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & -{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36}}{\int }_0^1{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & & ={\displaystyle \frac{3\aleph \left(\frac{5{\epsilon }_1+{\epsilon }_3}{6}\right)+3\aleph \left(\frac{{\epsilon }_1+5{\epsilon }_3}{6}\right)+2\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)}{8}}-{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{6^{1-\varphi }{({\epsilon }_3-{\epsilon }_1)}^{\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \times \left\{2^{1-\varphi }\left({}_{{\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)\right.\hfill \\ & & +2^{1-\varphi }\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}\right)\right)\hfill \\ \hfill & & \left.+\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}.\hfill \end{array}$$

•      For $\alpha =0,$$\beta ={\displaystyle \frac{13}{27}}$ and ${\epsilon }_2={\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}},$ identity (5) reduces to a post-quantum corrected Euler–Maclaurin-type identity:

$$\begin{array}{ccc}& & {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36}}{\int }_0^1{\displaystyle \frac{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\int }_0^1{\displaystyle \frac{39p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-80{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{80p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\hfill \\ & & \times {}_{{\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\int }_0^1{\displaystyle \frac{41p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-80{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{80p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\hfill \\ & & \times {}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma {\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & -{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36}}{\int }_0^1{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & & ={\displaystyle \frac{27\aleph \left(\frac{5{\epsilon }_1+{\epsilon }_3}{6}\right)+27\aleph \left(\frac{{\epsilon }_1+5{\epsilon }_3}{6}\right)+26\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)}{80}}-{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{6^{1-\varphi }{({\epsilon }_3-{\epsilon }_1)}^{\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \times \left\{2^{1-\varphi }\left({}_{{\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)\right.\hfill \\ & & +2^{1-\varphi }\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}\right)\right)\hfill \\ & & \left.+\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{5{\epsilon }_1+{\epsilon }_3}{6}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+5{\epsilon }_3}{6}}}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}.\hfill \end{array}$$

•      For $\beta =0,$$\alpha ={\displaystyle \frac{3}{8}}$ and ${\epsilon }_2={\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}},$ identity (5) reduces to a post quantum ${\displaystyle \frac{3}{8}}$-Simpson-type identity:

$$\begin{array}{ccc}& & {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\int }_0^1{\displaystyle \frac{5p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-8{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{8p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & -{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36}}{\int }_0^1{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}}+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36}}{\int }_0^1{\displaystyle \frac{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\hfill \\ & & \times {}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma {\displaystyle \frac{{\epsilon }_1+2{\epsilon }_3}{3}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\int }_0^1{\displaystyle \frac{3p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-8{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{8p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\displaystyle \frac{{\epsilon }_1+2{\epsilon }_3}{3}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+2{\epsilon }_3}{3}}+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & & ={\displaystyle \frac{\aleph \left({\epsilon }_1\right)+3\aleph \left(\frac{2{\epsilon }_1+{\epsilon }_3}{3}\right)+3\aleph \left(\frac{{\epsilon }_1+2{\epsilon }_3}{3}\right)+\aleph \left({\epsilon }_3\right)}{8}}-{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1)}{6^{1-\varphi }{({\epsilon }_3-{\epsilon }_1)}^{\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \times \left\{\left({}_{{\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+2{\epsilon }_3}{3}}\right)\right)\right.\hfill \\ & & \left.+2^{1-\varphi }\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}}\right)\right)+2^{1-\varphi }\left({}_{{\displaystyle \frac{{\epsilon }_1+2{\epsilon }_3}{3}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+2{\epsilon }_3}{3}}}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}.\hfill \end{array}$$

•      For $\alpha =0=\beta ,$ identity (5) reduces to a post-quantum companion Ostrowski-type identity:

$$\begin{array}{ccc}& & {\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1{\displaystyle \frac{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right){}_0{d}_{p,q}\sigma \hfill \\ & & -{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & +{\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}}{\int }_0^1{\displaystyle \frac{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}-{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\hfill \\ & & \times {}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right){}_0{d}_{p,q}\sigma \hfill \\ & & -{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}}{\int }_0^1{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & & ={\displaystyle \frac{\aleph \left({\epsilon }_2\right)+\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)}{2}}-{\displaystyle \frac{{\Gamma }_{p,q}(\varphi +1){({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1-\varphi }}{2^{1-\varphi }({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}-{\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^{1-\varphi }{\Gamma }_{p,q}(\varphi +1)}{({\epsilon }_3-{\epsilon }_1)p^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \times \left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}.\hfill \end{array}$$

To make the presentation concise, we adopt the following assumptions in our subsequent results.

$${\phi }_1:={\displaystyle \frac{{({\epsilon }_2-{\epsilon }_1)}^2}{{\epsilon }_3-{\epsilon }_1}},{\phi }_2:={\displaystyle \frac{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^2}{4({\epsilon }_3-{\epsilon }_1)}},\zeta :={\displaystyle \frac{\alpha }{1+\beta }},$$

$$\gamma :={\displaystyle \frac{\beta ({\epsilon }_3-{\epsilon }_1)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}},\delta :={\displaystyle \frac{{\epsilon }_3+{\epsilon }_1+2(\beta {\epsilon }_1-\beta {\epsilon }_2-{\epsilon }_2)}{(1+\beta )({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}},$$

$${}_p{a}_1^q:={\left|{\displaystyle \frac{{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)}{{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)}}\right|}^{\vartheta },{}_p{a}_2^q:={\left|{\displaystyle \frac{{}_{{\epsilon }_2}{D}_{p,q}\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)}{{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)}}\right|}^{\vartheta },{}_p{a}_3^q:={\left|{\displaystyle \frac{{}_{\frac{{\epsilon }_1+{\epsilon }_3}{2}}{D}_{p,q}\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)}{{}_{\frac{{\epsilon }_1+{\epsilon }_3}{2}}{D}_{p,q}\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)}}\right|}^{\vartheta },$$

$${}_p{a}_4^q:={\left|{\displaystyle \frac{{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)}{{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)}}\right|}^{\vartheta },$$

$${\left[{}_{\theta }^{\omega }{\mathfrak{A}}_{\phi ,p}^{\varphi }\right]}^{\theta }:={\int }_0^1{(1-\sigma )}^{\omega }{\left|\phi -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\theta }{}_0{d}_{p,q}\sigma ,$$

$${\left[{}_{\theta }^{\omega }{\mathfrak{B}}_{\phi ,p}^{\varphi }\right]}^{\theta }:={\int }_0^1{\sigma }^{\omega }{\left|\phi -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\theta }{}_0{d}_{p,q}\sigma ,$$

$${\left[{}_{\theta }^{a_i}{\mathfrak{C}}_{\phi ,p}^{\varphi }\right]}^{\theta }:={\int }_0^1{{}_p{a}_i^q}^{\sigma }{\left|\phi -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\theta }{}_0{d}_{p,q}\sigma ,$$

$${\mathcal{S}}_1^{\kappa }:={\int }_0^1\left|{\displaystyle \frac{45\sigma -14}{45}}\right|{\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{1157}{4050}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{90{\kappa }^{\frac{14}{45}}+(31\kappa -14)ln\kappa -45(1+\kappa )}{45{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

$${\mathcal{S}}_2^{\kappa }:={\int }_0^1\left|{\displaystyle \frac{15\sigma -11}{15}}\right|{\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{137}{450}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{30{\kappa }^{\frac{11}{15}}+(4\kappa -11)ln\kappa -15(1+\kappa )}{15{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

$${\mathcal{S}}_3^{\kappa }:={\int }_0^1\left|{\displaystyle \frac{15\sigma -4}{15}}\right|{\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{137}{450}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{30{\kappa }^{\frac{4}{15}}+(11\kappa -4)ln\kappa -15(1+\kappa )}{15{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

$${\mathcal{S}}_4^{\kappa }:={\int }_0^1\left|{\displaystyle \frac{45\sigma -31}{45}}\right|{\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{1157}{4050}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{90{\kappa }^{\frac{31}{45}}+(14\kappa -31)ln\kappa -45(1+\kappa )}{45{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

$${\mathcal{T}}_1^{\kappa }:={\int }_0^1\left|{\displaystyle \frac{8\sigma -3}{8}}\right|{\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{17}{64}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{16{\kappa }^{\frac{3}{8}}+(5\kappa -3)ln\kappa -8(1+\kappa )}{8{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

$${\mathcal{T}}_2^{\kappa }:={\int }_0^1|\sigma -1|{\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{\kappa -1-ln\kappa }{{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

$${\mathcal{T}}_3^{\kappa }:={\int }_0^1\sigma {\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{1-\kappa +\kappa ln\kappa }{{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

$${\mathcal{T}}_4^{\kappa }:={\int }_0^1\left|{\displaystyle \frac{8\sigma -5}{8}}\right|{\kappa }^{\sigma }d\sigma =\left\{\begin{array}{cc}{\displaystyle \frac{17}{64}},\hfill & \kappa =1,\hfill \\ {\displaystyle \frac{16{\kappa }^{\frac{5}{8}}+(3\kappa -5)ln\kappa -8(1+\kappa )}{8{(ln\kappa )}^2}},\hfill & \kappa \ne 1.\hfill \end{array}\right.$$

In our subsequent results, we have used the definition of s-convexity and log-convexity.

Definition 5  

([30]). The function $\aleph :[0,\infty )\to \mathbb{R}$ is said to be s-convex in the second sense if

$$\aleph (\sigma {\epsilon }_1+(1-\sigma ){\epsilon }_3)\le {\sigma }^s\aleph \left({\epsilon }_1\right)+{(1-\sigma )}^s\aleph \left({\epsilon }_3\right);\forall {\epsilon }_1,{\epsilon }_3\in [0,\infty ),\sigma \in [0,1],s\in (0,1].$$

Definition 6  

([31]). The function $\aleph :I\subset \mathbb{R}\to (0,\infty )$ is said to be log-convex if

$$\aleph (\sigma {\epsilon }_1+(1-\sigma ){\epsilon }_3)\le {[\aleph \left({\epsilon }_1\right)]}^{\sigma }{[\aleph \left({\epsilon }_3\right)]}^{1-\sigma };\forall {\epsilon }_1,{\epsilon }_3\in I,\sigma \in [0,1].$$

Theorem 1.  

Assume that conditions of Lemma 1 are satisfied for ${\epsilon }_1,{\epsilon }_3\in [0,\infty ).$ Further, if $|{}_{{\epsilon }_1}{D}_{p,q}\aleph |,$$|{}_{{\epsilon }_2}{D}_{p,q}\aleph |,$$\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|$ and $|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph |$ are s-convex functions on $({\epsilon }_1,{\epsilon }_3),$ then

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ \hfill & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \\ & & \le {\phi }_1\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|{}_1^s{\mathfrak{A}}_{1-\zeta ,p}^{\varphi }+{\phi }_1\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|{}_1^s{\mathfrak{B}}_{1-\zeta ,p}^{\varphi }+{\phi }_2\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|{}_1^s{\mathfrak{A}}_{\gamma ,p}^{\varphi }\hfill \\ & & +{\phi }_2\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_1^s{\mathfrak{B}}_{\gamma ,p}^{\varphi }+{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_1^s{\mathfrak{A}}_{\delta ,p}^{\varphi }\hfill \\ & & +{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|{}_1^s{\mathfrak{B}}_{\delta ,p}^{\varphi }+{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|{}_1^s{\mathfrak{A}}_{\zeta ,p}^{\varphi }\hfill \\ & & +{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|{}_1^s{\mathfrak{B}}_{\zeta ,p}^{\varphi }.\hfill \end{array}$$

(13)

Proof. 

By properties of modulus and s-convexities of $|{}_{{\epsilon }_1}{D}_{p,q}\aleph |,$$|{}_{{\epsilon }_2}{D}_{p,q}\aleph |,$$\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|$ and $|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph |,$ the following holds:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \\ & & \le {\phi }_1{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_1{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & \le {\phi }_1{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left\{{(1-\sigma )}^s|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)|+{\sigma }^s|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)|\right\}{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left\{{(1-\sigma )}^s|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)|+{\sigma }^s\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|\right\}{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left\{{(1-\sigma )}^s\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|\right.\hfill \\ & & \left.+{\sigma }^s|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)|\right\}{}_0{d}_{p,q}\sigma \hfill \\ \hfill & & +{\phi }_1{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left\{{(1-\sigma )}^s|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)|\right.\hfill \\ & & \left.+{\sigma }^s|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)|\right\}{}_0{d}_{p,q}\sigma \hfill \\ & & ={\phi }_1\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|{\int }_0^1{(1-\sigma )}^s\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_1\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|{\int }_0^1{\sigma }^s\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|{\int }_0^1{(1-\sigma )}^s\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{\int }_0^1{\sigma }^s\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{\int }_0^1{(1-\sigma )}^s\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|{\int }_0^1{\sigma }^s\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|{\int }_0^1{(1-\sigma )}^s\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|{\int }_0^1{\sigma }^s\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma .\hfill \end{array}$$

□

Corollary 1.  

For $p=1,$ in Theorem 1, the following Riemann–Liouville fractional quantum inequality holds:

$$\begin{array}{c}\\ & & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_q(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }}}\left\{\left({}_{{\epsilon }_1}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_2\right)\left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_3\right)\right\}\right.\hfill \\ & & -{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_q(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }}}\left.\left\{\left({}_{{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{J}_q^{\varphi }\aleph \right)({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right|\hfill \\ & & \le {\phi }_1\left|{}_{{\epsilon }_1}{D}_q\aleph \left({\epsilon }_1\right)\right|{}_1^s{\mathfrak{A}}_{1-\zeta ,1}^{\varphi }+{\phi }_1\left|{}_{{\epsilon }_1}{D}_q\aleph \left({\epsilon }_2\right)\right|{}_1^s{\mathfrak{B}}_{1-\zeta ,1}^{\varphi }+{\phi }_2\left|{}_{{\epsilon }_2}{D}_q\aleph \left({\epsilon }_2\right)\right|{}_1^s{\mathfrak{A}}_{\gamma ,1}^{\varphi }\hfill \\ & & +{\phi }_2\left|{}_{{\epsilon }_2}{D}_q\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_1^s{\mathfrak{B}}_{\gamma ,1}^{\varphi }+{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_1^s{\mathfrak{A}}_{\delta ,1}^{\varphi }\hfill \\ & & +{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|{}_1^s{\mathfrak{B}}_{\delta ,1}^{\varphi }+{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|{}_1^s{\mathfrak{A}}_{\zeta ,1}^{\varphi }\hfill \\ & & +{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left({\epsilon }_3\right)\right|{}_1^s{\mathfrak{B}}_{\zeta ,1}^{\varphi }.\hfill \end{array}$$

Theorem 2.  

Assume that conditions of Lemma 1 are satisfied for ${\epsilon }_1,{\epsilon }_3\in [0,\infty ).$ Further, if $|{}_{{\epsilon }_1}{D}_{p,q}{\aleph |}^{\mu },$$|{}_{{\epsilon }_2}{D}_{p,q}{\aleph |}^{\mu },$${\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|}^{\mu }$ and $|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}{\aleph |}^{\mu }$ are s-convex functions on $({\epsilon }_1,{\epsilon }_3),$ for $\mu >1$ such that $\vartheta ={\displaystyle \frac{\mu }{\mu -1}},$ then

$$\begin{array}{ccc}\hfill & & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \\ & & \le {\displaystyle \frac{{\phi }_1{}_{\vartheta }^0{\mathfrak{A}}_{1-\zeta ,p}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_{p,q}}}}\sqrt[\mu ]{{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|}^{\mu }+{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\mu }}\hfill \\ & & +{\displaystyle \frac{{\phi }_2{}_{\vartheta }^0{\mathfrak{A}}_{\gamma ,p}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_{p,q}}}}\sqrt[\mu ]{{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\mu }+{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }}\hfill \\ & & +{\displaystyle \frac{{\phi }_2{}_{\vartheta }^0{\mathfrak{A}}_{\delta ,p}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_{p,q}}}}\sqrt[\mu ]{{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }+{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }}\hfill \\ & & +{\displaystyle \frac{{\phi }_1{}_{\vartheta }^0{\mathfrak{A}}_{\zeta ,p}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_{p,q}}}}\sqrt[\mu ]{{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }+{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|}^{\mu }}.\hfill \end{array}$$

(14)

Proof. 

By properties of modulus, s-convexities of $|{}_{{\epsilon }_1}{D}_{p,q}{\aleph |}^{\mu },$$|{}_{{\epsilon }_2}{D}_{p,q}{\aleph |}^{\mu },$${\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|}^{\mu },$$|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}{\aleph |}^{\mu }$ and the Hölder’s inequality, the following holds:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \\ & & \le {\phi }_1{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_0{d}_{p,q}\sigma \hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & +{\phi }_2{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_1{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & \le {\phi }_1{\left[{\int }_0^1{\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}{\left[{\int }_0^1{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)\right|}^{\mu }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1{\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1{\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right|}^{\mu }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1{\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)\right|}^{\mu }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & & \le {\phi }_1{\left[{\int }_0^1{\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left\{{(1-\sigma )}^s{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|}^{\mu }+{\sigma }^s{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\mu }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1{\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left\{{(1-\sigma )}^s{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\mu }+{\sigma }^s{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1{\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left\{{(1-\sigma )}^s{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }+{\sigma }^s{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1{\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & \times {\left[{\int }_0^1\left\{{(1-\sigma )}^s{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }+{\sigma }^s{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|}^{\mu }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & ={\phi }_1{\left[{\int }_0^1{\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|}^{\mu }{\int }_0^1{(1-\sigma )}^s{}_0{d}_{p,q}\sigma +{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\mu }{\int }_0^1{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1{\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\mu }{\int }_0^1{(1-\sigma )}^s{}_0{d}_{p,q}\sigma +{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }{\int }_0^1{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1{\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\left[{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }{\int }_0^1{(1-\sigma )}^s{}_0{d}_{p,q}\sigma \right.\hfill \\ & & {\left.+{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }{\int }_0^1{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1{\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times \left[{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }{\int }_0^1{(1-\sigma )}^s{}_0{d}_{p,q}\sigma \right.{\left.+{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|}^{\mu }{\int }_0^1{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\mu }}}.\hfill \end{array}$$

□

Corollary 2.  

For $p=1,$ in Theorem 2, the following Riemann–Liouville fractional quantum inequality holds:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_q(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }}}\left\{\left({}_{{\epsilon }_1}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_2\right)\left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_3\right)\right\}\right.\hfill \\ & & -{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_q(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }}}\left.\left\{\left({}_{{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{J}_q^{\varphi }\aleph \right)({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right|\hfill \\ & & \le {\displaystyle \frac{{\phi }_1{}_{\vartheta }^0{\mathfrak{A}}_{1-\zeta ,1}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_q}}}\sqrt[\mu ]{{\left|{}_{{\epsilon }_1}{D}_q\aleph \left({\epsilon }_1\right)\right|}^{\mu }+{\left|{}_{{\epsilon }_1}{D}_q\aleph \left({\epsilon }_2\right)\right|}^{\mu }}\hfill \\ & & +{\displaystyle \frac{{\phi }_2{}_{\vartheta }^0{\mathfrak{A}}_{\gamma ,1}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_q}}}\sqrt[\mu ]{{\left|{}_{{\epsilon }_2}{D}_q\aleph \left({\epsilon }_2\right)\right|}^{\mu }+{\left|{}_{{\epsilon }_2}{D}_q\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }}\hfill \\ & & +{\displaystyle \frac{{\phi }_2{}_{\vartheta }^0{\mathfrak{A}}_{\delta ,1}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_q}}}\sqrt[\mu ]{{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\mu }+{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & & +{\displaystyle \frac{{\phi }_1{}_{\vartheta }^0{\mathfrak{A}}_{\zeta ,1}^{\varphi }}{\sqrt[\mu ]{{[s+1]}_q}}}\sqrt[\mu ]{{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\mu }+{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left({\epsilon }_3\right)\right|}^{\mu }}.\hfill \end{array}$$

Theorem 3.  

Assume that conditions of Lemma 1 are satisfied for ${\epsilon }_1,{\epsilon }_3\in [0,\infty ).$ Further, if $|{}_{{\epsilon }_1}{D}_{p,q}{\aleph |}^{\vartheta },$$|{}_{{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta },$${\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|}^{\vartheta }$ and $|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta }$ are s-convex functions on $({\epsilon }_1,{\epsilon }_3),$ for $\vartheta \ge 1,$ then

$$\begin{array}{ccc}\hfill & & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \\ & & \le {\phi }_1{\left[{}_1^0{\mathfrak{A}}_{1-\zeta ,p}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{1-\zeta ,p}^{\varphi }{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{1-\zeta ,p}^{\varphi }{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{}_1^0{\mathfrak{A}}_{\gamma ,p}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{\gamma ,p}^{\varphi }{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{\gamma ,p}^{\varphi }{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{}_1^0{\mathfrak{A}}_{\delta ,p}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{\delta ,p}^{\varphi }{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{\delta ,p}^{\varphi }{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1{\left[{}_1^0{\mathfrak{A}}_{\zeta ,p}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{\zeta ,p}^{\varphi }{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{\zeta ,p}^{\varphi }{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}.\hfill \end{array}$$

(15)

Proof. 

By properties of modulus, s-convexities of $|{}_{{\epsilon }_1}{D}_{p,q}{\aleph |}^{\vartheta },$$|{}_{{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta },$${\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|}^{\vartheta },$$|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta }$ and the power mean inequality, the following holds:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \end{array}$$

$$\begin{array}{ccc}& & \le {\phi }_1{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_1{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & \le {\phi }_1{\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \le {\phi }_1{\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left\{{(1-\sigma )}^s{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|}^{\vartheta }+{\sigma }^s{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\right.\hfill \end{array}$$

$$\begin{array}{ccc}& & {\left.\times \left\{{(1-\sigma )}^s{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }+{\sigma }^s{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\right.\hfill \\ & & {\left.\times \left\{{(1-\sigma )}^s{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }+{\sigma }^s{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\right.\hfill \\ & & \times {\left.\left\{{(1-\sigma )}^s{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }+{\sigma }^s{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|}^{\vartheta }\right\}{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & ={\phi }_1{\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times \left[{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|}^{\vartheta }{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{(1-\sigma )}^s{}_0{d}_{p,q}\sigma \right.\hfill \\ & & {\left.+{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times \left[{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{(1-\sigma )}^s{}_0{d}_{p,q}\sigma \right.\hfill \\ & & {\left.+{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times \left[{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{(1-\sigma )}^s{}_0{d}_{p,q}\sigma \right.\hfill \\ & & {\left.+{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times \left[{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{(1-\sigma )}^s{}_0{d}_{p,q}\sigma \right.\hfill \\ & & {\left.+{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|}^{\vartheta }{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\sigma }^s{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}.\hfill \end{array}$$

□

Corollary 3.  

For $p=1,$ in Theorem 3, the following Riemann–Liouville fractional quantum inequality holds:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_q(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }}}\left\{\left({}_{{\epsilon }_1}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_2\right)\left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\epsilon }_3\right)\right\}\right.\hfill \\ & & -{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_q(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }}}\left.\left\{\left({}_{{\epsilon }_2}{J}_q^{\varphi }\aleph \right)\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{J}_q^{\varphi }\aleph \right)({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right|\hfill \\ & & \le {\phi }_1{\left[{}_1^0{\mathfrak{A}}_{1-\zeta ,1}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{1-\zeta ,1}^{\varphi }{\left|{}_{{\epsilon }_1}{D}_q\aleph \left({\epsilon }_1\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{1-\zeta ,1}^{\varphi }{\left|{}_{{\epsilon }_1}{D}_q\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{}_1^0{\mathfrak{A}}_{\gamma ,1}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{\gamma ,1}^{\varphi }{\left|{}_{{\epsilon }_2}{D}_q\aleph \left({\epsilon }_2\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{\gamma ,1}^{\varphi }{\left|{}_{{\epsilon }_2}{D}_q\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{}_1^0{\mathfrak{A}}_{\delta ,1}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{\delta ,1}^{\varphi }{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{\delta ,1}^{\varphi }{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_q\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1{\left[{}_1^0{\mathfrak{A}}_{\zeta ,1}^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^s{\mathfrak{A}}_{\zeta ,1}^{\varphi }{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left({\epsilon }_1+{\epsilon }_3-{\epsilon }_2\right)\right|}^{\vartheta }+{}_1^s{\mathfrak{B}}_{\zeta ,1}^{\varphi }{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_q\aleph \left({\epsilon }_3\right)\right|}^{\vartheta }\right]}^{{\displaystyle \frac{1}{\vartheta }}}.\hfill \end{array}$$

Remark 3.  

For $\varphi =1$ and $q\to 1$, Theorems 2.1, 2.2 and 2.3 [29] are retrieved from Corollaries 1, 2 and 3, respectively. For different choices of parameters discussed in Remark 2, several error bounds of the well-known Newton–Cotes quadrature rules can be obtained involving at most five points.

Theorem 4.  

Assume that conditions of Lemma 1 are satisfied. Further, if $|{}_{{\epsilon }_1}{D}_{p,q}{\aleph |}^{\vartheta },$$|{}_{{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta },$${\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|}^{\vartheta }$ and $|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta }$ are log-convex functions on $({\epsilon }_1,{\epsilon }_3),$ for $\vartheta \ge 1,$ then

$$\begin{array}{ccc}\hfill & & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \\ & & \le {\phi }_1\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|{\left[{}_1^0{\mathfrak{A}}_{1-\zeta }^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^{a_1}{\mathfrak{C}}_{1-\zeta }^{\varphi }\right]}^{{\displaystyle \frac{1}{\vartheta }}}+{\phi }_2\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|{\left[{}_1^0{\mathfrak{A}}_{\gamma }^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^{a_2}{\mathfrak{C}}_{\gamma }^{\varphi }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{\left[{}_1^0{\mathfrak{A}}_{\delta }^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^{a_3}{\mathfrak{C}}_{\delta }^{\varphi }\right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right|{\left[{}_1^0{\mathfrak{A}}_{\zeta }^{\varphi }\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}{\left[{}_1^{a_4}{\mathfrak{C}}_{\zeta }^{\varphi }\right]}^{{\displaystyle \frac{1}{\vartheta }}}.\hfill \end{array}$$

(16)

Proof. 

By properties of modulus, log-convexities of $|{}_{{\epsilon }_1}{D}_{p,q}{\aleph |}^{\vartheta },$$|{}_{{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta },$${\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \right|}^{\vartheta },$$|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}{\aleph |}^{\vartheta }$ and the power mean inequality, the following holds:

$$\begin{array}{ccc}\hfill & & \left|{\displaystyle \frac{{\phi }_1(\zeta -1)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)-2{\phi }_2\delta ({\epsilon }_1-{\epsilon }_2)}{({\epsilon }_1-{\epsilon }_2)(2{\epsilon }_2-{\epsilon }_3-{\epsilon }_1)}}\left\{\aleph \left({\epsilon }_2\right)+\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right\}\right.\hfill \\ & & -{\displaystyle \frac{{\phi }_1\zeta }{{\epsilon }_1-{\epsilon }_2}}\left\{\aleph \left({\epsilon }_1\right)+\aleph \left({\epsilon }_3\right)\right\}+{\displaystyle \frac{\gamma ({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}{{\epsilon }_3-{\epsilon }_1}}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\hfill \\ & & -{\displaystyle \frac{{\phi }_1{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_2-{\epsilon }_1)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\left\{\left({}_{{\epsilon }_1}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1}{\Phi }_{p^{\varphi }}\left({\epsilon }_2\right)\right)\right.\hfill \\ & & \left.+\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\epsilon }_3\right)\right)\right\}-{\displaystyle \frac{2^{1+\varphi }{\phi }_2{\Gamma }_{p,q}(\varphi +1)}{{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)}^{1+\varphi }{p}^{\frac{{\varphi }^2+4\varphi -3}{2}}}}\hfill \\ & & \left.\times \left\{\left({}_{{\epsilon }_2}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\epsilon }_2}{\Phi }_{p^{\varphi }}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right)+\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{K}_{p,q}^{\varphi }\aleph \right)\left({}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{\Phi }_{p^{\varphi }}({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right\}\right|\hfill \\ & & \le {\phi }_1{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_2{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & +{\phi }_1{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)\right|{}_0{d}_{p,q}\sigma \hfill \\ & & \le {\phi }_1{\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ \hfill & & \times {\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right)\right|}^{\vartheta }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \le {\phi }_1{\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|}^{\vartheta (1-\sigma )}{\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta \sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|}^{\vartheta (1-\sigma )}{\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta \sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2{\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times \left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\right.\hfill \\ & & \times {\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|}^{\vartheta (1-\sigma )}{\left.{\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right|}^{\vartheta \sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1{\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times \left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|\right.\hfill \\ & & \times {\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right|}^{\vartheta (1-\sigma )}{\left.{\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_3\right)\right|}^{\vartheta \sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & ={\phi }_1\left|{}_{{\epsilon }_1}{D}_{p,q}\aleph \left({\epsilon }_1\right)\right|{\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{{}_p{a}_1^q}^{\sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2\left|{}_{{\epsilon }_2}{D}_{p,q}\aleph \left({\epsilon }_2\right)\right|{\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{{}_p{a}_2^q}^{\sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_2\left|{}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|{\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{{}_p{a}_3^q}^{\sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & +{\phi }_1\left|{}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right|{\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{}_0{d}_{p,q}\sigma \right]}^{1-{\displaystyle \frac{1}{\vartheta }}}\hfill \\ & & \times {\left[{\int }_0^1\left|\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right|{{}_p{a}_4^q}^{\sigma }{}_0{d}_{p,q}\sigma \right]}^{{\displaystyle \frac{1}{\vartheta }}}.\hfill \end{array}$$

□

Corollary 4.  

For $\alpha ={\displaystyle \frac{14}{39}},$$\beta ={\displaystyle \frac{2}{13}},$${\epsilon }_2={\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}},$$p=1=\varphi ,$ and $q\to 1,$ in Theorem 4, the following inequality in classical calculus holds:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{7\aleph \left({\epsilon }_1\right)+32\aleph \left(\frac{3{\epsilon }_1+{\epsilon }_3}{4}\right)+12\aleph \left(\frac{{\epsilon }_1+{\epsilon }_3}{2}\right)+32\aleph \left(\frac{{\epsilon }_1+3{\epsilon }_3}{4}\right)+7\aleph \left({\epsilon }_3\right)}{90}}-{\displaystyle \frac{1}{{\epsilon }_3-{\epsilon }_1}}{\int }_{{\epsilon }_1}^{{\epsilon }_3}\aleph \left(\sigma \right)d\sigma \right|\hfill \\ & & \le {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{16}}{\left[{\displaystyle \frac{1157}{4050}}\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}\left\{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|\sqrt[\vartheta ]{{\mathcal{S}}_1^{{}_1{a}_1^{1^{-}}}}+\left|{\aleph }^{\prime }\left({\displaystyle \frac{{\epsilon }_1+3{\epsilon }_3}{4}}\right)\right|\sqrt[\vartheta ]{{\mathcal{S}}_4^{{}_1{a}_4^{1^{-}}}}\right\}\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{16}}{\left[{\displaystyle \frac{137}{450}}\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}\left\{\left|{\aleph }^{\prime }\left({\displaystyle \frac{3{\epsilon }_1+{\epsilon }_3}{4}}\right)\right|\sqrt[\vartheta ]{{\mathcal{S}}_2^{{}_1{a}_2^{1^{-}}}}+\left|{\aleph }^{\prime }\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|\sqrt[\vartheta ]{{\mathcal{S}}_3^{{}_1{a}_3^{1^{-}}}}\right\}\hfill \\ & & \le {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{16}}{\left[{\displaystyle \frac{1157}{4050}}\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}\left\{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|\sqrt[\vartheta ]{{\mathcal{S}}_1^{{}_1{a}_1^{1^{-}}}}+\sqrt[4]{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|}\sqrt[4]{{\left|{\aleph }^{\prime }\left({\epsilon }_3\right)\right|}^3}\sqrt[\vartheta ]{{\mathcal{S}}_4^{{}_1{a}_4^{1^{-}}}}\right\}\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{16}}{\left[{\displaystyle \frac{137}{450}}\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}\left\{\sqrt[4]{\left|{\aleph }^{\prime }\left({\epsilon }_3\right)\right|}\sqrt[4]{{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|}^3}\sqrt[\vartheta ]{{\mathcal{S}}_2^{{}_1{a}_2^{1^{-}}}}+\sqrt{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|}\sqrt{\left|{\aleph }^{\prime }\left({\epsilon }_3\right)\right|}\sqrt[\vartheta ]{{\mathcal{S}}_3^{{}_1{a}_3^{1^{-}}}}\right\}.\hfill \end{array}$$

Corollary 5.  

For $\alpha ={\displaystyle \frac{3}{8}},$ $\beta =0,$${\epsilon }_2={\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}},$$p=1=\varphi ,$ and $q\to 1,$ in Theorem 4, the following inequality in classical calculus holds:

$$\begin{array}{ccc}& & \left|{\displaystyle \frac{\aleph \left({\epsilon }_1\right)+3\aleph \left(\frac{2{\epsilon }_1+{\epsilon }_3}{3}\right)+3\aleph \left(\frac{{\epsilon }_1+2{\epsilon }_3}{3}\right)+\aleph \left({\epsilon }_3\right)}{8}}-{\displaystyle \frac{1}{{\epsilon }_3-{\epsilon }_1}}{\int }_{{\epsilon }_1}^{{\epsilon }_3}\aleph \left(\sigma \right)d\sigma \right|\hfill \\ & & \le {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\left[{\displaystyle \frac{17}{64}}\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}\left\{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|\sqrt[\vartheta ]{{\mathcal{T}}_1^{{}_1{a}_1^{1^{-}}}}+\left|{\aleph }^{\prime }\left({\displaystyle \frac{{\epsilon }_1+2{\epsilon }_3}{3}}\right)\right|\sqrt[\vartheta ]{{\mathcal{T}}_4^{{}_1{a}_4^{1^{-}}}}\right\}\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36{\left(2\right)}^{\frac{\vartheta -1}{\vartheta }}}}\left\{\left|{\aleph }^{\prime }\left({\displaystyle \frac{2{\epsilon }_1+{\epsilon }_3}{3}}\right)\right|\sqrt[\vartheta ]{{\mathcal{T}}_2^{{}_1{a}_2^{1^{-}}}}+\left|{\aleph }^{\prime }\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)\right|\sqrt[\vartheta ]{{\mathcal{T}}_3^{{}_1{a}_3^{1^{-}}}}\right\}\hfill \\ & & \le {\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{9}}{\left[{\displaystyle \frac{17}{64}}\right]}^{{\displaystyle \frac{\vartheta -1}{\vartheta }}}\left\{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|\sqrt[\vartheta ]{{\mathcal{T}}_1^{{}_1{a}_1^{1^{-}}}}+\sqrt[3]{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|}\sqrt[3]{{\left|{\aleph }^{\prime }\left({\epsilon }_3\right)\right|}^2}\sqrt[\vartheta ]{{\mathcal{T}}_4^{{}_1{a}_4^{1^{-}}}}\right\}\hfill \\ & & +{\displaystyle \frac{{\epsilon }_3-{\epsilon }_1}{36{\left(2\right)}^{\frac{\vartheta -1}{\vartheta }}}}\left\{\sqrt[3]{\left|{\aleph }^{\prime }\left({\epsilon }_3\right)\right|}\sqrt[3]{{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|}^2}\sqrt[\vartheta ]{{\mathcal{T}}_2^{{}_1{a}_2^{1^{-}}}}+\sqrt{\left|{\aleph }^{\prime }\left({\epsilon }_1\right)\right|}\sqrt{\left|{\aleph }^{\prime }\left({\epsilon }_3\right)\right|}\sqrt[\vartheta ]{{\mathcal{T}}_3^{{}_1{a}_3^{1^{-}}}}\right\}.\hfill \end{array}$$

4. Numerical and Graphical Analysis

This section presents the key findings of our study both graphically and numerically, making it easier to comprehend the theoretical conclusions. The figures and tables in each example are not related to each other. The statistical sets were selected at random. For example, when q and p are close and far apart, we have attempted to take all the conceivable values in each selection. In each example, we first explain the parameters in a specific table and then the parameters in a specific graph by mentioning the table and graph. In our computation, we determine the values independently for the left and right sides of each pertinent inequality of the related theorem. For the purposes of this section, let $\aleph \left(\sigma \right)={\sigma }^2$.

Example 1.  

The inequality (13) of Theorem 1 is numerically analyzed in Table 1, where $\varphi =1,{\epsilon }_1=10,{\epsilon }_2=16.7,{\epsilon }_3=29,$ and $q=0.0001.$ Assuming $0.001\le p\le 1,0\le \beta \le 1,\alpha =0.99,$$s=0.22,$ Figure 1 offers a graphic representation of the validity of inequality (13) with the same values for $\varphi ,{\epsilon }_1,{\epsilon }_2,{\epsilon }_3,$ and $q.$

Table 1. Statistical analysis of Theorem 1.

p	$\mathit{\alpha }$	$\mathit{\beta }$	s	LHS of (13)	RHS of (13)	RHS-LHS	RHS/LHS
0.1	0.99	0.1	1	155.3487	965.0222	809.6735	6.21197
0.3	0.8	0.2	0.7	151.2356	325.0792	173.8436	2.1495
0.5	0.6	0.5	0.4	147.0268	275.2382	128.2114	1.8720
0.7	0.3	0.9	0.2	143.1058	291.3878	148.2820	2.0362
0.9	0.001	1	0.1	140.0466	304.3630	164.3164	2.1733

Figure 1. LHS and RHS of (13) with $0.001\le p\le 1,$ and $0\le \beta \le 1$.

Example 2.  

The inequality (14) of Theorem 2 is numerically analyzed in Table 2, where $\varphi =2=\mu ,{\epsilon }_1=-10,{\epsilon }_2=-9,$ and ${\epsilon }_3=10.$ Assuming $0.0001\le q\le 0.999,0\le \alpha \le 1,p=1$, $\beta =0.001,s=0.01,$ Figure 2 offers a graphic representation of the validity of inequality (14) with the same values for $\varphi ,\mu ,{\epsilon }_1,{\epsilon }_2,$ and ${\epsilon }_3.$

Table 2. Statistical analysis of Theorem 2.

q	p	$\mathit{\alpha }$	$\mathit{\beta }$	s	LHS of (14)	RHS of (14)	RHS-LHS	RHS/LHS
0.01	0.2	0	1	0.0001	5.0166	39.0305	34.0139	7.7803
0.3	0.4	0.7	0.5	0.2	21.1556	32.9505	11.7949	1.5575
0.5	0.6	1	0.8	0.4	12.8871	36.4225	23.5354	2.8263
0.7	0.8	0.4	0.3	0.6	29.6569	39.4718	9.8149	1.3309
0.99	1	0.9	0	1	50.1877	61.4645	11.2768	1.2247

Figure 2. LHS and RHS of (14) with $0.0001\le q\le 0.999,$ and $0\le \alpha \le 1.$.

Example 3.  

The inequality (15) of Theorem 3 is numerically analyzed in Table 3, where $\varphi =1,{\epsilon }_1=20,{\epsilon }_2=22,{\epsilon }_3=25,$ and $q=0.0001.$ Assuming $0.001\le p\le 1,0\le \alpha \le 1,\beta =1,\vartheta =2,s=0.01,$ Figure 3 offers a graphic representation of the validity of inequality (15) with the same values for $\varphi ,{\epsilon }_1,{\epsilon }_2,{\epsilon }_3,$ and $q.$

Table 3. Statistical analysis of Theorem 3.

p	$\mathit{\alpha }$	$\mathit{\beta }$	s	$\mathit{\vartheta }$	LHS of (15)	RHS of (15)	RHS-LHS	RHS/LHS
0.01	0.9	0.1	0.99	1	64.7627	6.1718 $\times {10}^3$	6107.0373	95.2987
0.2	0.7	0.99	0.2	1.5	51.4046	85.7732	34.3686	1.6686
0.4	0.5	1	0.5	2	48.3309	78.4894	30.1585	1.6240
0.8	0.3	0.3	0.01	3	47.6885	48.2251	0.5366	1.0112
1	0.1	0.7	0.8	4	42.5085	52.4363	9.9278	1.2335

Figure 3. LHS and RHS of (15) with $0.001\le p\le 1,$ and $0\le \alpha \le 1.$.

Discussion on the Tightness of the Bounds

In addition to verifying that the left-hand side is less than or equal to the right-hand side of the obtained inequalities in theorems for the chosen parameters, we also examine the gap between these two sides to assess the informativeness of the bounds. For each example, we compute both the absolute difference $RHS-LHS$ and the relative ratio $RHS/LHS.$ The results show that for most parameter choices, the ratio lies between 1 and 2, indicating reasonably tight bounds. However, at extreme parameter values such as $q=0.0001$ and $p=0.01$, the ratio becomes larger, i.e., 95.2987. This behavior is expected, as such extreme parameters lie far from the classical regime and represent boundary cases in the post-quantum framework. The primary contribution of this work remains the unification and generalization of integral inequalities within post-quantum calculus, and the numerical examples serve to validate the analytical results across the parameter domain.

5. Application to Special Means

1.

Arithmetic mean

$$\mathsf{A}({\epsilon }_1,{\epsilon }_3)={\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}},{\epsilon }_1,{\epsilon }_3\in (-\infty ,\infty ).$$

2.

Geometric mean

$$\mathsf{G}({\epsilon }_1,{\epsilon }_3)=\sqrt{{\epsilon }_1{\epsilon }_3},{\epsilon }_1,{\epsilon }_3\in [0,\infty ).$$

Proposition 1.  

Let ${\epsilon }_1,{\epsilon }_3\in [0,\infty )$ such that ${\epsilon }_1<{\epsilon }_3,$ then

$$\begin{array}{ccc}& & \left|\mathsf{A}\left(\left({\epsilon }_2+{\epsilon }_1+2({\epsilon }_3-{\epsilon }_2)\zeta \right){\phi }_1,\left((\gamma +\delta +2)\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right)+{\epsilon }_2(\gamma -\delta )\right){\phi }_2\right)\right.\hfill \\ & & -{\displaystyle \frac{p^4{\left[2\right]}_{p,q}}{{\left[3\right]}_{p,q}{\left[4\right]}_{p,q}}}\mathsf{A}\left(2({\epsilon }_2-{\epsilon }_1){\phi }_1,({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2){\phi }_2\right)\hfill \\ & & \left.-{\displaystyle \frac{p^2}{{\left[3\right]}_{p,q}}}\mathsf{A}\left(2(2{\epsilon }_1+{\epsilon }_3-{\epsilon }_2){\phi }_1,({\epsilon }_1+{\epsilon }_3+2{\epsilon }_2){\phi }_2\right)\right|\hfill \\ & & \le {\displaystyle \frac{{\phi }_1}{\sqrt{{[s+1]}_{p,q}}}}\left[\mathsf{A}\left({(1-\zeta )}^2,p^2\right)-{\mathsf{G}}^2(1-\zeta ,p)+\mathsf{A}\left({\displaystyle \frac{q^6}{{\left[5\right]}_{p,q}}},{\displaystyle \frac{q^2{(q+p)}^2}{{\left[3\right]}_{p,q}}}\right)\right.\hfill \\ & & {\left.+(1-\zeta -p)\left\{{\mathsf{G}}^2\left({\displaystyle \frac{q}{{\left[2\right]}_{p,q}}},q+p\right)-{\mathsf{G}}^2\left({\displaystyle \frac{1}{{\left[3\right]}_{p,q}}},q^3\right)\right\}-{\mathsf{G}}^2\left({\displaystyle \frac{q^4}{{\left[4\right]}_{p,q}}},q+p\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & & \times {\left(8{{\epsilon }_1}^2+{\displaystyle \frac{{\left[2\right]}_{p,q}}{2}}({\epsilon }_2-{\epsilon }_1)\left({\left[2\right]}_{p,q}({\epsilon }_2-{\epsilon }_1)+4{\epsilon }_1\right)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & & +{\displaystyle \frac{{\phi }_2}{\sqrt{{[s+1]}_{p,q}}}}\left[\mathsf{A}\left({\gamma }^2,p^2\right)-{\mathsf{G}}^2(\gamma ,p)+\mathsf{A}\left({\displaystyle \frac{q^6}{{\left[5\right]}_{p,q}}},{\displaystyle \frac{q^2{(q+p)}^2}{{\left[3\right]}_{p,q}}}\right)\right.\hfill \\ & & {\left.+(\gamma -p)\left\{{\mathsf{G}}^2\left({\displaystyle \frac{q}{{\left[2\right]}_{p,q}}},q+p\right)-{\mathsf{G}}^2\left({\displaystyle \frac{1}{{\left[3\right]}_{p,q}}},q^3\right)\right\}-{\mathsf{G}}^2\left({\displaystyle \frac{q^4}{{\left[4\right]}_{p,q}}},q+p\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & & \times {\left(4{\epsilon }_2^2+{\displaystyle \frac{{\left[2\right]}_{p,q}}{4}}({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)\left({\left[2\right]}_{p,q}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2}}\right)+4{\epsilon }_2\right)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & & +{\displaystyle \frac{{\phi }_2}{\sqrt{{[s+1]}_{p,q}}}}\left[\mathsf{A}\left({\delta }^2,p^2\right)-{\mathsf{G}}^2(\delta ,p)+\mathsf{A}\left({\displaystyle \frac{q^6}{{\left[5\right]}_{p,q}}},{\displaystyle \frac{q^2{(q+p)}^2}{{\left[3\right]}_{p,q}}}\right)\right.\hfill \\ & & {\left.+(\delta -p)\left\{{\mathsf{G}}^2\left({\displaystyle \frac{q}{{\left[2\right]}_{p,q}}},q+p\right)-{\mathsf{G}}^2\left({\displaystyle \frac{1}{{\left[3\right]}_{p,q}}},q^3\right)\right\}-{\mathsf{G}}^2\left({\displaystyle \frac{q^4}{{\left[4\right]}_{p,q}}},q+p\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & & \times {\left({({\epsilon }_1+{\epsilon }_3)}^2+{\displaystyle \frac{{\left[2\right]}_{p,q}}{4}}({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)\left({\left[2\right]}_{p,q}\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2}}\right)+2({\epsilon }_1+{\epsilon }_3)\right)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & & +{\displaystyle \frac{{\phi }_1}{\sqrt{{[s+1]}_{p,q}}}}\left[\mathsf{A}\left({\zeta }^2,p^2\right)-{\mathsf{G}}^2(\zeta ,p)+\mathsf{A}\left({\displaystyle \frac{q^6}{{\left[5\right]}_{p,q}}},{\displaystyle \frac{q^2{(q+p)}^2}{{\left[3\right]}_{p,q}}}\right)\right.\hfill \\ & & {\left.+(\zeta -p)\left\{{\mathsf{G}}^2\left({\displaystyle \frac{q}{{\left[2\right]}_{p,q}}},q+p\right)-{\mathsf{G}}^2\left({\displaystyle \frac{1}{{\left[3\right]}_{p,q}}},q^3\right)\right\}-{\mathsf{G}}^2\left({\displaystyle \frac{q^4}{{\left[4\right]}_{p,q}}},q+p\right)\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & & \times {\left(4{({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)}^2+{\displaystyle \frac{{\left[2\right]}_{p,q}}{2}}({\epsilon }_2-{\epsilon }_1)\left({\left[2\right]}_{p,q}({\epsilon }_2-{\epsilon }_1)+4({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right)\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

Proof. 

Let $\aleph \left(\sigma \right)={\sigma }^2$ in Theorem 2, with $\varphi =2=\vartheta ,$ then

$${}_{{\epsilon }_1}{D}_{p,q}\aleph \left(\sigma \right)={\left[2\right]}_{p,q}(\sigma -{\epsilon }_1)+2{\epsilon }_1,$$

$${}_2^0{\mathfrak{A}}_{\phi ,p}^2={(\phi -p)}^2+{\displaystyle \frac{q^2{(q+p)}^2}{{\left[3\right]}_{p,q}}}+{\displaystyle \frac{q^6}{{\left[5\right]}_{p,q}}}+{\displaystyle \frac{2q(\phi -p)(q+p)}{{\left[2\right]}_{p,q}}}-{\displaystyle \frac{2q^3(\phi -p)}{{\left[3\right]}_{p,q}}}-{\displaystyle \frac{2q^4(q+p)}{{\left[4\right]}_{p,q}}},$$

$$\begin{array}{ccc}& & {\int }_0^1\left(1-\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right){}_{{\epsilon }_1}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_1+\sigma {\epsilon }_2\right){}_0{d}_{p,q}\sigma \hfill \\ & & =({\epsilon }_2-{\epsilon }_1)(1-\zeta )+2{\epsilon }_1(1-\zeta )-{\displaystyle \frac{({\epsilon }_2-{\epsilon }_1)p^4{\left[2\right]}_{p,q}}{{\left[3\right]}_{p,q}{\left[4\right]}_{p,q}}}-{\displaystyle \frac{2{\epsilon }_1{p}^2}{{\left[3\right]}_{p,q}}},\hfill \end{array}$$

$$\begin{array}{ccc}& & {\int }_0^1\left(\gamma -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right){}_{{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma ){\epsilon }_2+\sigma {\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}\right){}_0{d}_{p,q}\sigma \hfill \\ & & =\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2}}\right)\gamma +2{\epsilon }_2\gamma -{\displaystyle \frac{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)p^4{\left[2\right]}_{p,q}}{2{\left[3\right]}_{p,q}{\left[4\right]}_{p,q}}}-{\displaystyle \frac{2{\epsilon }_2{p}^2}{{\left[3\right]}_{p,q}}},\hfill \end{array}$$

$$\begin{array}{ccc}& & {\int }_0^1\left(\delta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right){}_{{\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}}{D}_{p,q}\aleph \left((1-\sigma ){\displaystyle \frac{{\epsilon }_1+{\epsilon }_3}{2}}+\sigma ({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\right){}_0{d}_{p,q}\sigma \hfill \\ & & =\left({\displaystyle \frac{{\epsilon }_1+{\epsilon }_3-2{\epsilon }_2}{2}}\right)\delta +({\epsilon }_1+{\epsilon }_3)\delta -{\displaystyle \frac{({\epsilon }_1+{\epsilon }_3-2{\epsilon }_2)p^4{\left[2\right]}_{p,q}}{2{\left[3\right]}_{p,q}{\left[4\right]}_{p,q}}}-{\displaystyle \frac{({\epsilon }_1+{\epsilon }_3)p^2}{{\left[3\right]}_{p,q}}},\hfill \\ & & {\int }_0^1\left(\zeta -{\displaystyle \frac{{(1-{}_0{\Phi }_q\left(\sigma \right))}_{p,q}^{\left(\varphi \right)}}{p^{\left(\genfrac{}{}{0pt}{}{\varphi }{2}\right)}}}\right){}_{{\epsilon }_1+{\epsilon }_3-{\epsilon }_2}{D}_{p,q}\aleph \left((1-\sigma )({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)+\sigma {\epsilon }_3\right){}_0{d}_{p,q}\sigma \hfill \\ & & =({\epsilon }_2-{\epsilon }_1)\zeta +2({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)\zeta -{\displaystyle \frac{({\epsilon }_2-{\epsilon }_1)p^4{\left[2\right]}_{p,q}}{{\left[3\right]}_{p,q}{\left[4\right]}_{p,q}}}-{\displaystyle \frac{2({\epsilon }_1+{\epsilon }_3-{\epsilon }_2)p^2}{{\left[3\right]}_{p,q}}}.\hfill \end{array}$$

Substituting the previously given values in inequality (14) of Theorem 2 yields the required result. □

Remark 4.  

The derivation of the inequality obtained in Proposition 1 involving the arithmetic and geometric means is not a mere substitution. Expressing the results in terms of special means is nontrivial, particularly when dealing with multiparameters, q-shifting operators, s-convexity, $(p,q)$-derivative, and the Riemann–Liouville fractional $(p,q){}_{{\epsilon }_1}$-integrals. Each condition must be carefully verified, and the parameters must be appropriately selected to ensure compatibility with the domain of the special means. This application serves to illustrate that the general theoretical framework developed in this paper is capable of producing concrete inequalities involving classical functions, thereby demonstrating its practical reach beyond abstract theory.

6. Conclusions

In this article, we identified a broad class of new post-quantum multiparameter integral inequalities involving the Riemann–Liouville fractional integral, successfully unifying the Boole-type, Bullen–Simpson-type, Maclaurin-type, corrected Euler–Maclaurin-type, ${\displaystyle \frac{3}{8}}$-Simpson-type, and companion Ostrowski-type frameworks. Using a variety of analytical tools, including s-convexity, logarithmic convexity, properties of the modulus, the Holder inequality, and the power mean inequality, we have established bounds that are both rigorous and adaptable to various function behaviors. Such theoretical developments are supported by meticulously crafted numerical and graphical examples, which show the accuracy and usefulness of the obtained estimates. In addition, the use of these inequalities in special means theory establishes a concrete connection between post-quantum fractional analysis and the classical mean values. The results given here add to the ever-increasing literature on post-quantum fractional integral inequalities while offering scope for future research in related areas under generalized convexity assumptions.