Evolutionary Approach to Inequalities of Hermite–Hadamard–Mercer Type for Generalized Wright’s Functions Associated with Computational Evaluation and Their Applications

Abstract

The theory of integral inequalities has a wide range of applications in physics and numerical computation, and plays a fundamental role in mathematical analysis. The present study delves into the attractive domain of Hermite–Hadamard–Mercer (H–H–M)-type inequalities having a special emphasis on Wright’s general functions, referred to as Raina’s functions in the scientific literature. The main goal of our progressive study is to use Raina’s Fractional Integrals to derive two useful lemmas for second-differentiable functions. Using the derived lemmas, we proved a large number of fractional integral inequalities related to trapezoidal and midpoint-type inequalities where those that are twice differentiable in absolute values are convex. Some of these results also generalize findings from previous research. Next, we provide applications to error estimates for trapezoidal and midpoint quadrature formulas and to analytical evaluations involving modified Bessel functions of the first kind and q-digamma functions, and we show the validity of the proposed inequalities in numerical integration and analysis of special functions. Finally, the results are well-supported by numerous examples, including graphical representations and numerical tables, which collectively highlight their accuracy and computational significance.

1. Introduction

Fractional calculus is a branch of mathematical analysis that focuses on integrals and derivatives of fractional orders other than natural numbers or integers. Fractional calculus is quite popular among researchers due to its behavior and applications in various scientific fields, not only in mathematical sciences, but also in bioengineering [1], epidemiology [2], physics [3], nanotechnology [4] and control systems [5]. Apart from its uses in these fields, fractional calculus has significantly advanced inequality theory, especially in the development of fractional inequalities. Mathematicians have examined many kinds of fractional integral inequalities in great depth over the last ten years. A wide range of fractional integral inequalities have been constructed by researchers using fractional calculus because of the significance of these in approximation theory. Based on their fundamental properties, researchers discovered novel fractional operators and applied them to a variety of real-world problems.

Many of the novel ideas and techniques arising from the convexity have a basic role in the development of almost all the fields of pure and also applied sciences, primarily in operators research, finance, computer science and engineering. It was proven that there exists a close relationship between the theory of convexity and integral inequalities (II), emphasizing their importance in the frame of fractional analysis.

It will be considered $x_n\ge x_{n-1}\ge \cdots \ge x_2\ge x_1>0$ and let $\lambda =({\lambda }_1,{\lambda }_2,\dots {\lambda }_n)$ nonnegative weights with ${\sum }_{i=1}^n{\lambda }_i=1$. The Jensen inequality [6,7] says that if f is a CF on $[{\nu }_1,{\nu }_2]$; then we obtain the following:

$$f\left({\sum }_{i=1}^n{\lambda }_i{x}_i\right)\le {\sum }_{i=1}^n{\lambda }_{j}f\left(x_i\right),$$

where for all $x_i\in [{\nu }_1,{\nu }_2]$ and ${\lambda }_i\in [0,1]$, $(i=\overline{1,n})$.

For a CF $f:I\subseteq R\to R$ on I with ${\nu }_1,{\nu }_2\in I$ and ${\nu }_1<{\nu }_2$ the H–H inequality states the following [8]:

$$f\left({\displaystyle \frac{{\nu }_1+{\nu }_2}{2}}\right)\le {\displaystyle \frac{1}{{\nu }_2-{\nu }_1}}{\int }_{{\nu }_1}^{{\nu }_2}f\left(x\right)dx\le {\displaystyle \frac{f\left({\nu }_1\right)+f\left({\nu }_2\right)}{2}}.$$

(1)

The H–H inequality (1) is one of the most well-known and applied inequalities, published by Hermite [9] in 1883 and by Hadamard [8] in 1893.

Some applications to special means of real numbers and trapezoidal formulas can be found in [10,11].

Theorem 1

(See [12]). “If f is a convex function on $I=[{\nu }_1,{\nu }_2]$, then the J–M inequality states the following:

$$f\left({\nu }_1+{\nu }_2-{\sum }_{i=1}^n{\lambda }_i{x}_i\right)\le f\left({\nu }_1\right)+f\left({\nu }_2\right)-{\sum }_{i=1}^n{\lambda }_{i}f\left(x_i\right),$$

for each $x_i\in [{\nu }_1,{\nu }_2]$ and ${\lambda }_i\in [0,1]$, $(i=\overline{1,n})$ with ${\sum }_{i=1}^n{\lambda }_i=1$.”

The definitions used in this study will be provided using the previously indicated essential convex function inequalities.

Definition 1

([13]). “Let $f\in L[{\nu }_1,{\nu }_2]$. The left-sided and right-sided R-L fractional integrals ${\mathcal{J}}_{\vartheta ,{\nu }_1^{+}}\left(f\right)\left(x\right)$ and ${\mathcal{J}}_{\vartheta ,{\nu }_2^{-}}\left(f\right)\left(x\right)$ of order $\vartheta >0$ with ${\nu }_1\ge 0$ can be defined, respectively, by the following:

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{\vartheta ,{\nu }_1^{+}}\left(f\right)\left(x\right)& ={\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_{{\nu }_1}^x{(x-\eta )}^{\vartheta -1}f\left(\eta \right)d\eta ,\hfill & \hfill x>{\nu }_1,\end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathcal{J}}_{\vartheta ,{\nu }_2^{-}}\left(f\right)\left(x\right)& ={\displaystyle \frac{1}{\Gamma \left(\vartheta \right)}}{\int }_x^{{\nu }_2}{(\eta -x)}^{\vartheta -1}f\left(\eta \right)d\eta ,\hfill & \hfill x<{\nu }_2,\end{array}$$

where Γ is a Euler Gamma function and defined as follows: $\Gamma \left(\vartheta \right)={\int }_0^{\infty }{e}^{-\eta }{\eta }^{\vartheta -1}d\eta .$”

It is important to mention here, in this point the remarkable work of Wright, about hypergeometric and related higher transcedental functions [14,15,16], where he studied the asymptotic expansion of the Taylor–MacLaurin from below:

$$\begin{array}{c}\hfill {\mathcal{E}}_{\alpha ,\beta }(\mathsf{\Phi },z)=\sum _{n=0}^{\infty }{\displaystyle \frac{\mathsf{\Phi }\left(n\right)}{\Gamma (\alpha n+\beta )}}{z}^n,\end{array}$$

where $\alpha ,\beta \in \mathbb{C},\Re \left(\alpha \right)>0,$ and $\mathsf{\Phi }\left(t\right)$ is a function which satisfy suitable conditions. Earlier reports detailed developments of simpler cases of this formula in 1905 by Mittag–Leffler, Wiman in 1905, E.W. Barnes in 1906, G.H. Hardy in 1905, G. N., Watson in 1913, Fox in 1928 and others.

This function was considered recently in [17], which was related to fractional calculus, and a survey article on recent developments on higher transcedental functions of analytic number theory can be found in [18].

Raina in [17] introduced a new class [19] of functions defined by

$${\mathcal{F}}_{\rho ,\vartheta }^{\sigma }\left(x\right)=\sum _{i=0}^{\infty }{\displaystyle \frac{\sigma \left(i\right)}{\Gamma (\rho i+\vartheta )}}{x}^i$$

$\rho >0$, $\vartheta >0;$ with $|x|<R$ where $\sigma \left(i\right)$ is a bounded sequence of positive real numbers and R is the set of real numbers. The above class of function is the generalization of classical Mittag–Leffler function and the Kummer function. If $\rho =1$, $\vartheta =0$ and $\sigma \left(i\right)={\displaystyle \frac{{\left({\theta }_1\right)}_i{\left({\theta }_2\right)}_i}{{\left({\theta }_3\right)}_i}}$ for $i=1,2,3,\dots$ where ${\theta }_1,$${\theta }_2$ and ${\theta }_3$ are parameters which can take arbitrary real or complex values (provided that ${\theta }_3\ne 0,-1,-2,\dots$), and the symbol ${\left({\theta }_1\right)}_i$ denotes the quantity

$$\begin{array}{c}\hfill {\left({\theta }_1\right)}_i={\displaystyle \frac{\Gamma ({\theta }_1+i)}{\Gamma \left({\theta }_1\right)}}={\theta }_1({\theta }_1-1)({\theta }_1-2)\dots ({\theta }_1+i-1),i=0,1,2,\dots ,\end{array}$$

and restricts its domain to $|z|>1$ (with $z\in C$), then we have the classical hypergeometric function given in [20], that is

$$\begin{array}{c}\hfill {}_{2}F_3({\theta }_1,{\theta }_2;{\theta }_3;z)=\sum _{i=1}^{\infty }{\displaystyle \frac{{\left({\theta }_1\right)}_i{\left({\theta }_2\right)}_i}{i!{\left({\theta }_3\right)}_i}}{z}^i.\end{array}$$

Moreover, if $\sigma =(1,1,1,\dots )$ with $\rho ={\theta }_1$, $Re\left({\theta }_1\right)>0$, $\vartheta =1$, then

$$\begin{array}{c}\hfill E{}_{{\theta }_1}\left(z\right)=\sum _{i=0}^{\infty }{\displaystyle \frac{z^i}{\Gamma (1+{\theta }_{1}i)}},\end{array}$$

The above exact function, which intermittently appears in the investigation of fractional integrals and derivatives is called a classical Mittag–Leffler function, and was first considered by Magnus Gustaf (Gösta) Mittag–Leffler (1846–1927) in 1903 and Anders Wiman (1865–1959) in 1905 (see, for details, [21,22,23]).

Definition 2

([17,24]). “Left-sided and right-sided Raina’s fractional integrals are respectively defined as follows:

$${\mathcal{J}}_{\rho ,\vartheta ,{\nu }_1^{+};w}^{\sigma }\left(f\right)\left(x\right)={\int }_{{\nu }_1}^x{(x-\eta )}^{\vartheta -1}{\mathcal{F}}_{\rho ,\vartheta }^{\sigma }\left[w{(x-\eta )}^{\rho }\right]f\left(\eta \right)d\eta ,(x>{\nu }_1>0),$$

and

$${\mathcal{J}}_{\rho ,\vartheta ,{{\nu }_2}^{-};w}^{\sigma }\left(f\right)\left(x\right)={\int }_x^{{\nu }_2}{(\eta -x)}^{\vartheta -1}{\mathcal{F}}_{\rho ,\vartheta }^{\sigma }\left[w{(\eta -x)}^{\rho }\right]f\left(\eta \right)d\eta ,(0<x<{\nu }_2),$$

$\vartheta ,\rho >0,w\in R$ and $f\left(\eta \right)$ is such that the integral exists. In addition it is known that ${\mathcal{J}}_{\rho ,\vartheta ,{{\nu }_1}^{+};w}^{\sigma }\left(f\right)\left(x\right)$ and ${\mathcal{J}}_{\rho ,\vartheta ,{{\nu }_2}^{-};w}^{\sigma }\left(f\right)\left(x\right)$ are bounded integral operators on $L({\nu }_1,{\nu }_2)$ if

$$\mathcal{M}={\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{({\nu }_2-{\nu }_1)}^{\rho }\right]<\infty .\mathit{\text{”}}$$

Recently, Set E. et al. [25] introduced some H–H–M-type integral inequalities for functions whose first derivative of absolute value is convex using Raina’s fractional integral. Here, we give two of their lemmas for our new development.

Lemma 1

([25]). Let $f:[{\nu }_1,{\nu }_2]\to R$ be a differentiable mapping on $({\nu }_1,{\nu }_2),$ with ${\nu }_1<{\nu }_2$, $\vartheta >0.$ If $f^{\prime }\in L[{\nu }_1,{\nu }_2]$ then following identity for generalized fractional integral operators holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ \hfill & ={\displaystyle \frac{z_2-z_1}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}[{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right)d\eta \hfill \\ & -{\int }_0^1{(1-\eta )}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right)d\eta ],\hfill \end{array}$$

(2)

for all $z_1,z_2\in [{\nu }_1,{\nu }_2]$, $\rho , \vartheta >0,$$w\in \mathcal{R}$ and $\eta \in [0,1].$

Lemma 2

([25]). Let $f:[{\nu }_1,{\nu }_2]\to R$ be a differentiable mapping on $({\nu }_1,{\nu }_2),$ with ${\nu }_1<{\nu }_2$, $\vartheta >0.$ If $f^{\prime }\in L[{\nu }_1,{\nu }_2]$ then following identity for generalized fractional integral operators holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)\hfill \\ \hfill & ={\displaystyle \frac{z_2-z_1}{4}}[{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)d\eta \hfill \\ & -{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right)d\eta ],\hfill \end{array}$$

(3)

for all $z_1,z_2\in [{\nu }_1,{\nu }_2]$, $\rho , \vartheta >0,$$w\in \mathcal{R}$ and $\eta \in [0,1].$

In the recent results obtained by [26], several inequalities (midpoint, trapezoid, Milne, and Simpson-type inequalities) were obtained using diverse classes of functions, like convex functions, bounded functions, Lipschitzian functions, and functions of bounded variation for the case of two real parameters for three times differentiable functions.

H–H–type inequalities, in fact trapezoidal-type inequalities in the Hilbert spaces, can be found, for example, in [27].

In [28,29], researchers were provided some newly H–H–M-type inequalities for twice-differentiable CF which extended certain pre-existing inequalities in the literature. Last year H–H–M-type inequalities for fractional integrals in the case of functions whose absolute values of the third derivatives are convex have been established in [30], and in [31] new different trapezoid and midpoint H–H–M-type inequalities have been provided for fractional integrals and fractional integral operators in the case of functions whose absolute values of the second and third derivatives are convex. In [32] it was analyzed what would become the generalized H–H–M inequality by using ${\psi }_k$-Raina’s fractional integrals (${\psi }_k$-RFI). Many applications for special functions like the modified Bessel function and q-digamma function were presented.

Motivated by findings from [28,30,31] and [29,33,34,35], we introduced a number of integral inequalities related to the left- and right-hand sides of H–H–M-type inequalities in the framework of the Raina’s fractional integral operator for the second differentiable functions instead (gap in the field).

In order to motivate this study, we first sought to determine what the inequalities from [28], Theorem 3, Theorem 4, and [29], Theorem 3.3, Theorem 3.10, as well as from [34], Lemma 2.1, would be in the case of twice differentiable convex functions in the framework of Raina’s fractional integrals.

The main advantage of our work is the ability to offer previously established results for particular setting of the parameters while also improving and generalizing many of the existing results in the subject of integral inequalities. For example, Lemma 3 extends Lemma 2.1 in [34], Lemma 1 in [35] and Lemma 3.1 in [29], whereas Lemma 4 extends Lemma 1 in [28] and Lemma 2 in [36] specifically. Additionally, Theorem 2 applies to Theorem 2.2, for $m=1$ of [34], Theorems 3.3 and 3.10 of [29] and Remark 3.11 of [29]. Both Theorem 3 and Theorem 4 generalize Theorem 2.4, for $m=1$ of [34] and Theorem 5 of [28], respectively. Moreover, Theorem 6 generalizes Theorem 4 of [28], while Theorem 5 extends Theorem 5 of [28].

Furthermore, the study of inequalities is essential to developing theoretical and practical applications in mathematical analysis and optimization. Applications of established inequalities can enhance the precision and effectiveness of optimization algorithms, especially those that incorporate convexity assumptions. These findings lead to more efficient solutions for operations research resource allocation problems by offering better bounds and estimations.

The results can be applied to the development of inequalities associated with optimization problems in economics and finance. Convex functions are frequently employed in financial models to depict resource allocation as well as utility function, production function, and cost functions. In probability theory and statistics, where convex functions are commonly used in relation to probability distributions, cumulative distribution functions, and moments of random variables, these inequalities can also be useful. For example, they can be helpful when evaluating the statistical properties of random variables.

Recently discovered integral-based inequalities and convex optimization can significantly improve machine learning and optimization algorithms. The behavior of objective functions and constraints in convex optimization problems can be studied with help of these inequalities. For physicists and engineers working with fractional-order models, the resultant inequalities can be helpful, particularly in domains like control systems and signal processing. Our findings extend on certain applications of modified Bessel and q-digamma functions that were introduced in [37], including Examples 1, 2, and 3. Furthermore, special means of real numbers have a wide range of applications [38]. The applicability of our findings, see Proposition 3, Proposition 4, Proposition 5 and Proposition 6, where a similar variety of inequalities are derived from [37] (Examples 1 and 3) and [10].

A significant aspect of the Casimir theory of dielectric spheres is modified Bessel functions in [37,39,40,41]. Potential applications of the addressed inequalities in the above case can also be explored. Moreover, as demonstrated in [42], Example 4.2 deals with modified Bessel functions of the second kind, whereas Example 4.3 (page 12) addresses q-digamma functions, for which comparable novel developments can be studied. The Sabaheh approach can be utilized to examine our findings in the context of Example 4.4 from [42], page 13.

In Section 1, the necessary concepts and results which will be used below are recalled; then, in Section 2, two trapezoidal-type inequalities for twice differentiable CF via Raina’s fractional integrals are given in Theorems 2 and 3 after the first auxiliary identity given in Lemma 3, which is used in their demonstrations. In Section 3, a new identity is provided by Lemma 4 and then three midpoint-type inequalities for twice differentiable CF via Raina’s fractional integrals are obtained in Theorems 4–6. Section 4 is dedicated to applications to numerical trapezoidal and midpoint quadrature formulas with Propositions 1 and 2, to modified Bessel functions in Propositions 3 and 4 and q-diqamma function Propositions 5 and 6. Several particular cases are examinated in Section 5 with graphical representations which validate our results in Examples 1–4. In Section 6 provides conclusions and future research.

2. Trapezoidal Type Inequalities for Twice Differentiable Convex Functions via Raina’s Fractional Integrals

Here, we give trapezoid type for twice differential convex functions via Raina’s Fractional Integrals. We need the following lemma for obtaining the following results:

Lemma 3.

Let $f:[{\nu }_1,{\nu }_2]\to R$ be a twice differentiable mapping on $({\nu }_1,{\nu }_2),$ with ${\nu }_1<{\nu }_2$, $\vartheta >0.$ If $f^{″}\in L[{\nu }_1,{\nu }_2]$ then following identity for generalized fractional integral operators holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ \hfill & ={\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}{\int }_0^1[{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]-{(1-\eta )}^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]\hfill \\ & -{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]]f^{″}\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right)d\eta ,\hfill \end{array}$$

(4)

for all $z_1,z_2\in [{\nu }_1,{\nu }_2]$, $\rho , \vartheta >0,$$w\in \mathcal{R}$ and $\eta \in [0,1].$

Proof. 

By using integration by parts, we get

$$\begin{array}{cc}\hfill & {\int }_0^1[{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]-{(1-\eta )}^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]\hfill \\ & -{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]]f^{″}\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right)d\eta \hfill \\ & ={\displaystyle \frac{{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]-{(1-\eta )}^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]-{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]}{z_2-z_1}}\hfill \\ & \times f^{\prime }\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right){|}_0^1+{\displaystyle \frac{1}{z_2-z_1}}{\int }_0^1[{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]\hfill \\ & -{(1-\eta )}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]]f^{\prime }\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right)d\eta \hfill \\ & ={\displaystyle \frac{1}{z_2-z_1}}{\int }_0^1\left[{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]-{(1-\eta )}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]\right]\hfill \\ & \times f^{\prime }\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right)d\eta ,\hfill \end{array}$$

(5)

The intended identity in (4) arisen from (5) by considering (2) and rearranging the terms. □

Remark 1.

If we put $\sigma \left(0\right)=1,$$j=0$ and $\vartheta =1,$ in Lemma 3, then Lemma 3 gives [29] [Lemma 3.1].

Remark 2.

If we take $\sigma \left(0\right)=1,$$j=0,$$z_1={\nu }_1$ and $z_2={\nu }_2$, in Lemma 3, then Lemma 3 gives [34] [Lemma 2.1].

Remark 3.

If we take $\sigma \left(0\right)=1,$$j=0,$$z_1={\nu }_1$, $z_2={\nu }_2$ and $\vartheta =1$, in Lemma 3, then Lemma 3 gives [35] [Lemma 1].

Theorem 2.

Assume that $f:[{\nu }_1,{\nu }_2]\to R$ be a twice differentiable function on $({\nu }_1,{\nu }_2)$ with ${\nu }_1<{\nu }_2$. If $|f^{″}{|}^q$, is CF on $[{\nu }_1,{\nu }_2]$, $q\ge 1,$ then next inequality for generalized fractional integrals holds:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}{\mathcal{F}}_{\rho ,\vartheta +3}^{{\sigma }_1}\left[w{\left(z_2-z_1\right)}^{\rho }\right]{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|f^{″}\left(z_1\right){|}^q+{|f^{″}\left(z_2\right)|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\sigma }_1\left(j\right)& =(\rho j+\vartheta )\sigma \left(j\right),\hfill \end{array}$$

for all $z_1, z_2\in [{\nu }_1,{\nu }_2],$$z_1<z_2,$$\rho , \vartheta >0$ and $w\in \mathcal{R}.$

Proof. 

From Lemma 3, by using the inequality of power-mean, we get

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\left({\int }_0^1\right\{{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]-{(1-\eta )}^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]\hfill \\ & -{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]\}d\eta {)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times \left({\int }_0^1\right\{{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]-{(1-\eta )}^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]\hfill \\ & -{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]\}|f^{″}\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right){|}^{q}d\eta {)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left(z_2-z_1\right)}^{\rho j}{\left({\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)d\eta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times {\left({\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)|f^{″}\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \end{array}$$

Using the inequality of J–M because of the convexity of $|f^{″}{|}^q$, we obtain

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left(z_2-z_1\right)}^{\rho j}{\left({\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)d\eta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times {\left({\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-\left(\eta |f^{″}\left(z_1\right){|}^q+(1-\eta ){|f^{″}\left(z_2\right)|}^q\right)\right)d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left(z_2-z_1\right)}^{\rho j}{\left({\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)d\eta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times (\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q\right){\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)d\eta \hfill \\ & -{\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)\left(\eta |f^{″}\left(z_1\right){|}^q+(1-\eta ){|f^{″}\left(z_2\right)|}^q\right)d\eta {)}^{{\displaystyle \frac{1}{q}}}\hfill \end{array}$$

The desired result follows from previous inequality and by using next computations:

$${\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)d\eta ={\displaystyle \frac{\rho j+\vartheta }{\rho j+\vartheta +2}},$$

$${\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)\eta d\eta ={\int }_0^1\left(1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}\right)(1-\eta )d\eta ={\displaystyle \frac{\rho j+\vartheta }{2(\rho j+\vartheta +2)}},$$

this finishes the proof. □

Remark 4.

If we put $\sigma \left(0\right)=1,$$j=0$ and $\vartheta =q=1,$ in Theorem 2, then Theorem 2 gives [29] [Theorem 3.3].

Remark 5.

If we take $\sigma \left(0\right)=1,$$j=0$ and $\vartheta =1,$ in Theorem 2, then Theorem 2 gives [29] [Theorem 3.10].

Remark 6.

If we take $\sigma \left(0\right)=1,$$j=0,$$z_1={\nu }_1$ and $z_2={\nu }_2$, in Theorem 2, then Theorem 2 gives [34] [Theorem 2.2, for m = 1].

Remark 7.

If we take $\sigma \left(0\right)=1,$$j=0,$$z_1={\nu }_1$, $z_2={\nu }_2$ and $\vartheta =1$, in Theorem 2, then Theorem 2 gives [29] [Remark 3.11].

Theorem 3.

We assume that $f:[{\nu }_1,{\nu }_2]\to R$ is a twice differentiable function on $({\nu }_1,{\nu }_2)$ with $f^{″}\in L[{\nu }_1,{\nu }_2],$ and ${\nu }_1<{\nu }_2$. If $|f^{″}{|}^q$ is CF on $[{\nu }_1,{\nu }_2]$, when $1<q$, then the generalized fractional integral inequality holds:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}{\mathcal{F}}_{\rho ,\vartheta +2}^{{\sigma }_2}\left[w{\left(z_2-z_1\right)}^{\rho }\right]{\left({\displaystyle |}{f}^{″}\left({\nu }_1\right){{\displaystyle |}}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|f^{″}\left(z_1\right){|}^q+{|f^{″}\left(z_2\right)|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}},\hfill \end{array}$$

where

$${\sigma }_2\left(j\right)={\left({\displaystyle \frac{(\rho j+\vartheta +1)p-1}{(\rho j+\vartheta +1)p+1}}\right)}^{{\displaystyle \frac{1}{p}}}\sigma \left(j\right),$$

for all $z_1, z_2\in [{\nu }_1,{\nu }_2],$$z_1<z_2,$${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,$$\rho >0$, $\vartheta >0$ and, $w\in \mathcal{R}.$

Proof. 

By Lemma 3, and by using the Hölder’s inequality, we obtain

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\left({\int }_0^1\right|{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]-{(1-\eta )}^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{(1-\eta )}^{\rho }\right]\hfill \\ & -{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\eta }^{\rho }\right]{|}^{p}d\eta {)}^{{\displaystyle \frac{1}{p}}}{\left({\int }_0^1|f^{″}\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left(z_2-z_1\right)}^{\rho j}{\left({\int }_0^1|1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}{|}^{p}d\eta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \times {\left({\int }_0^1|f^{″}\left({\nu }_1+{\nu }_2-\left(\eta z_1+(1-\eta )z_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Using J–M inequality because $|f^{″}{|}^q$ is a CF, we get

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{f({\nu }_1+{\nu }_2-z_2)+f({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{1}{2{(z_2-z_1)}^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\hfill \\ & \times \left[\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_2)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{({\nu }_1+{\nu }_2-z_1)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left(z_2-z_1\right)}^{\rho j}{\left({\int }_0^1|1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}{|}^{p}d\eta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \times {\left({\int }_0^1\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-\left(\eta |f^{″}\left(z_1\right){|}^q+(1-\eta ){|f^{″}\left(z_2\right)|}^q\right)\right)d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & ={\displaystyle \frac{{(z_2-z_1)}^2}{2{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\right]}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left(z_2-z_1\right)}^{\rho j}{\left({\int }_0^1|1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}{|}^{p}d\eta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \times {\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|f^{″}\left(z_1\right){|}^q+{|f^{″}\left(z_2\right)|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

In order to evaluate the integral ${\int }_0^1|1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}{|}^{p}d\eta$, we see that for any $A>B\ge 0$ and $p\ge 1$, it will be obtained ${(A-B)}^p\le A^p-B^p.$ Therefore, we get that

$$\begin{array}{c}\hfill |1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}{|}^p\le 1-{(1-\eta )}^{(\vartheta +1+\rho j)p}-{\eta }^{(\vartheta +1+\rho j)p},\end{array}$$

for all $\eta \in [0,1]$. Thus we find that

$$\begin{array}{cc}\hfill {\int }_0^1|1-{(1-\eta )}^{\vartheta +1+\rho j}-{\eta }^{\vartheta +1+\rho j}{|}^{p}d\eta & \le {\int }_0^1\left(1-{(1-\eta )}^{(\vartheta +1+\rho j)p}-{\eta }^{(\vartheta +1+\rho j)p}\right)d\eta \hfill \\ & ={\displaystyle \frac{(\rho j+\vartheta +1)p-1}{(\rho j+\vartheta +1)p+1}}.\hfill \end{array}$$

This finishes the demonstration of the Theorem 3. □

Remark 8.

If we take $\sigma \left(0\right)=1,$$j=0,$$z_1={\nu }_1$ and $z_2={\nu }_2$, in Theorem 3, then Theorem 3 gives [34] [Theorem 2.4, for m = 1].

Remark 9.

If we consider $\sigma \left(0\right)=1,$$j=0,$$z_1={\nu }_1$, $z_2={\nu }_2$ and $\vartheta =1$, in Theorem 3, then we have

$$\begin{array}{c}\hfill |{\displaystyle \frac{f\left({\nu }_1\right)+f\left({\nu }_2\right)}{2}}-{\displaystyle \frac{1}{{\nu }_2-{\nu }_1}}{\int }_{{\nu }_1}^{{\nu }_2}f\left(x\right)dx|\le {\displaystyle \frac{{({\nu }_2-{\nu }_1)}^2}{4}}{\left({\displaystyle \frac{2p-1}{2p+1}}\right)}^{{\displaystyle \frac{1}{p}}}{\left({\displaystyle \frac{|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}}.\end{array}$$

3. Midpoint Type Inequalities for Twice Differentiable Convex Functions via Raina’s Fractional Integrals

Here we give Midpoint type for twice differential convex functions via Raina’s Fractional Integrals. We need the following lemma for obtaining those results:

Lemma 4.

Let $f:[{\nu }_1,{\nu }_2]\to R$ be a twice differentiable mapping on $({\nu }_1,{\nu }_2),$ with ${\nu }_1<{\nu }_2$, $\vartheta >0.$ If $f^{″}\in L[{\nu }_1,{\nu }_2]$ then following identity for generalized fractional integral operators holds:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)\hfill \\ \hfill & ={\displaystyle \frac{{(z_2-z_1)}^2}{8}}[{\int }_0^1{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)d\eta \hfill \\ & +{\int }_0^1{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right)d\eta ],\hfill \end{array}$$

(6)

for all $z_1,z_2\in [{\nu }_1,{\nu }_2]$, $\rho ,\vartheta >0,$$w\in \mathcal{R}$ and $\eta \in [0,1].$

Proof. 

It will be denoted by $T_1$ and $T_2$ next integrals:

$$\begin{array}{c}\hfill T_1={\int }_0^1{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)d\eta ,\end{array}$$

and

$$\begin{array}{c}\hfill T_2={\int }_0^1{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right)d\eta .\end{array}$$

By integrating by parts $T_1$, we successively get

$$\begin{array}{cc}\hfill T_1& =-{\displaystyle \frac{2{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left(\frac{\eta }{2}\right)}^{\rho }\right]}{z_2-z_1}}{f}^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right){|}_0^1\hfill \\ & +{\displaystyle \frac{2}{z_2-z_1}}{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)d\eta \hfill \\ & =-{\displaystyle \frac{2{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\frac{1}{2^{\rho }}\right]}{z_2-z_1}}{f}^{\prime }\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_2+z_1}{2}}\right)\hfill \\ & +{\displaystyle \frac{2}{z_2-z_1}}{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)d\eta ,\hfill \end{array}$$

By a similar argument, one gets

$$\begin{array}{cc}\hfill T_2& ={\displaystyle \frac{2{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }\frac{1}{2^{\rho }}\right]}{z_2-z_1}}{f}^{\prime }\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_2+z_1}{2}}\right)\hfill \\ & -{\displaystyle \frac{2}{z_2-z_1}}{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right)d\eta .\hfill \end{array}$$

We can write

$$\begin{array}{cc}\hfill T_1+T_2& ={\displaystyle \frac{2}{z_2-z_1}}[{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)d\eta \hfill \\ & -{\int }_0^1{\eta }^{\vartheta }{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]f^{\prime }\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right)d\eta ].\hfill \end{array}$$

(7)

The desired identity (6) follows from (7) if we use (3) and if we rearrange the terms. □

Remark 10.

If we consider $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1$, in Lemma 4, then Lemma 4 gives [28] [Lemma 1].

Remark 11.

If we take $\sigma \left(0\right)=1,$$j=0,$$z_1={\nu }_1$, $z_2={\nu }_2$ and $\vartheta =1$, in Lemma 4, then Lemma 4 gives [36] [Lemma 2].

Theorem 4.

We assume that $f:[{\nu }_1,{\nu }_2]\to R$ be a twice differentiable function on $({\nu }_1,{\nu }_2)$ with ${\nu }_1<{\nu }_2$. If $|f^{″}|$ is CF on $[{\nu }_1,{\nu }_2],$ then next identity takes place:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{4}}{\mathcal{F}}_{\rho ,\vartheta +3}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]\left[|f^{″}\left({\nu }_1\right)|+|f^{″}\left({\nu }_2\right)|-{\displaystyle \frac{|f^{″}\left(z_1\right)|+|f^{″}\left(z_2\right)|}{2}}\right],\hfill \end{array}$$

for all $z_1, z_2\in [{\nu }_1,{\nu }_2],$$z_1<z_2,$$\rho , \vartheta >0$, and $w\in \mathcal{R}.$

Proof. 

By using the Lemma 4, properties of modulus, we obtain

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\left[{\int }_0^1\right|{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]\left|\right|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)|d\eta \hfill \\ & +{\int }_0^1\left|{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]\right|\left|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right)\right|d\eta ]\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)\left|w\right|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j{\int }_0^1{\eta }^{\vartheta +1+\rho j}\left[\right|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right)|\hfill \\ & +\left|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right)\right|]d\eta .\hfill \end{array}$$

Using J–M inequality because $|f^{″}|$ is a CF, we get

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j[2\left(|f^{″}\left({\nu }_1\right)|+|f^{″}\left({\nu }_2\right)|\right){\int }_0^1{\eta }^{\rho j+\vartheta +1}d\eta \hfill \end{array}$$

$$\begin{array}{cc}\hfill & -\left(|f^{″}\left(z_1\right)|+|f^{″}\left(z_2\right)|\right){\int }_0^1{\eta }^{\rho j+\vartheta +1}d\eta ]\hfill \\ & ={\displaystyle \frac{{(z_2-z_1)}^2}{4}}{\mathcal{F}}_{\rho ,\vartheta +3}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]\left[|f^{″}\left({\nu }_1\right)|+|f^{″}\left({\nu }_2\right)|-{\displaystyle \frac{|f^{″}\left(z_1\right)|+|f^{″}\left(z_2\right)|}{2}}\right],\hfill \end{array}$$

which completes the proof. □

Remark 12.

If we take $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1$, in Theorem 4, then Theorem 4 gives [28] [Theorem 3].

Theorem 5.

We suppose that $f:[{\nu }_1,{\nu }_2]\to R$ be a twice differentiable function on $({\nu }_1,{\nu }_2)$ such that $f^{″}\in L[{\nu }_1,{\nu }_2],$${\nu }_1<{\nu }_2$. If $|f^{″}{|}^q$ is a CF on $[{\nu }_1,{\nu }_2]$, when $1<q$, then the generalized fractional integral inequality holds:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}{\mathcal{F}}_{\rho ,\vartheta +2}^{{\sigma }_3}\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right][{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{3|f^{″}\left(z_1\right){|}^q+{|f^{″}\left(z_2\right)|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|f^{″}\left(z_1\right){|}^q+3{|f^{″}\left(z_2\right)|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}],\hfill \end{array}$$

where

$${\sigma }_3\left(j\right)={\left({\displaystyle \frac{1}{(\rho j+\vartheta +1)p+1}}\right)}^{{\displaystyle \frac{1}{p}}}\sigma \left(j\right),$$

for all $z_1, z_2\in [{\nu }_1,{\nu }_2],$$z_1<z_2,$${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1,$$\rho , \vartheta >0$ and $w\in \mathcal{R}.$

Proof. 

From Lemma 4, by utilizing the Hölder’s inequality, we find

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}{\left({\int }_0^1|{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]{|}^{p}d\eta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \times \{{\left({\int }_0^1|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left({\int }_0^1|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\}\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j{\left({\int }_0^1|{\eta }^{\vartheta +1+\rho j}{|}^{p}d\eta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \times \{{\left({\int }_0^1|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left({\int }_0^1|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\}.\hfill \end{array}$$

Using J–M inequality because $|f^{″}{|}^q$ is a CF, we obtain

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j{\left({\int }_0^1|{\eta }^{\vartheta +1+\rho j}{|}^{p}d\eta \right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \times \{{\left({\int }_0^1\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-\left({\displaystyle \frac{2-\eta }{2}}|f^{″}\left(z_1\right){|}^q+{\displaystyle \frac{\eta }{2}}{|f^{″}\left(z_2\right)|}^q\right)\right)d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left({\int }_0^1\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-\left({\displaystyle \frac{\eta }{2}}|f^{″}\left(z_1\right){|}^q+{\displaystyle \frac{2-\eta }{2}}{|f^{″}\left(z_2\right)|}^q\right)\right)d\eta \right)}^{{\displaystyle \frac{1}{q}}}\}\hfill \\ & ={\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j{\left({\displaystyle \frac{1}{(\rho j+\vartheta +1)p+1}}\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \times [{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{3|f^{″}\left(z_1\right){|}^q+{|f^{″}\left(z_2\right)|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|f^{″}\left(z_1\right){|}^q+3{|f^{″}\left(z_2\right)|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}]\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}{\mathcal{F}}_{\rho ,\vartheta +2}^{{\sigma }_3}\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right][{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{3|f^{″}\left(z_1\right){|}^q+{|f^{″}\left(z_2\right)|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|f^{″}\left(z_1\right){|}^q+3{|f^{″}\left(z_2\right)|}^q}{4}}\right)}^{{\displaystyle \frac{1}{q}}}].\hfill \end{array}$$

Thus the proof of the Theorem 5 is completed. □

Remark 13.

If we put $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1$, in Theorem 5, then Theorem 5 gives [28] [Theorem 5].

Theorem 6.

We suppose that $f:[{\nu }_1,{\nu }_2]\to R$ is a twice differentiable function on $({\nu }_1,{\nu }_2)$ and ${\nu }_1<{\nu }_2$. If $|f^{″}{|}^q$, is a CF on $[{\nu }_1,{\nu }_2]$, $1\le q,$ then next inequality for generalized fractional integrals holds:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}{\mathcal{F}}_{\rho ,\vartheta +3}^{{\sigma }_4}\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right],\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\sigma }_4\left(j\right)=& [{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{(\rho j+\vartheta +4)|f^{″}\left(z_1\right){|}^q+(\rho j+\vartheta +2){|f^{″}\left(z_2\right)|}^q}{2(\rho j+\vartheta +3)}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{(\rho j+\vartheta +2)|f^{″}\left(z_1\right){|}^q+(\rho j+\vartheta +4){|f^{″}\left(z_2\right)|}^q}{2(\rho j+\vartheta +3)}}\right)}^{{\displaystyle \frac{1}{q}}}]\sigma \left(j\right)\hfill \end{array}$$

for all $z_1, z_2\in [{\nu }_1,{\nu }_2],$$z_1<z_2,$$\rho , \vartheta >0$ and $w\in \mathcal{R}.$

Proof. 

By Lemma 4, and by utilizing the power-mean inequality, we get

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}{\left({\int }_0^1{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]d\eta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times \{{\left({\int }_0^1{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left({\int }_0^1{\eta }^{\vartheta +1}{\mathcal{F}}_{\rho ,\vartheta +2}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\left({\displaystyle \frac{\eta }{2}}\right)}^{\rho }\right]|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\}\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j{\left({\int }_0^1{\eta }^{\vartheta +1+\rho j}d\eta \right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times \{{\left({\int }_0^1{\eta }^{\vartheta +1+\rho j}|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{2-\eta }{2}}{z}_1+{\displaystyle \frac{\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left({\int }_0^1{\eta }^{\vartheta +1+\rho j}|f^{″}\left({\nu }_1+{\nu }_2-\left({\displaystyle \frac{\eta }{2}}{z}_1+{\displaystyle \frac{2-\eta }{2}}{z}_2\right)\right){|}^{q}d\eta \right)}^{{\displaystyle \frac{1}{q}}}\}.\hfill \end{array}$$

Using J–M inequality because $|f^{″}{|}^q$ is a CF, we have

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{2^{\vartheta -1}}{{(z_2-z_1)}^{\vartheta }}}\left[\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{+};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_1)+\left({\mathcal{J}}_{\rho ,\vartheta ,{\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)}^{-};w}^{\sigma }f\right)({\nu }_1+{\nu }_2-z_2)\right]\hfill \\ & -{\mathcal{F}}_{\rho ,\vartheta +1}^{\sigma }\left[w{\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right]f\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j{\left({\displaystyle \frac{1}{\rho j+\vartheta +2}}\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times \{{\left({\int }_0^1{\eta }^{\vartheta +1+\rho j}\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-\left({\displaystyle \frac{2-\eta }{2}}|f^{″}\left(z_1\right){|}^q+{\displaystyle \frac{\eta }{2}}{|f^{″}\left(z_2\right)|}^q\right)\right)d\eta \right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\int }_0^1{\eta }^{\vartheta +1+\rho j}\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-\left({\displaystyle \frac{\eta }{2}}|f^{″}\left(z_1\right){|}^q+{\displaystyle \frac{2-\eta }{2}}{|f^{″}\left(z_2\right)|}^q\right)\right)d\eta \right)}^{{\displaystyle \frac{1}{q}}}\}\hfill \\ \hfill & \le {\displaystyle \frac{{(z_2-z_1)}^2}{8}}\sum _{j=0}^{\infty }{\displaystyle \frac{{\sigma \left(j\right)|w|}^j}{\Gamma (\rho j+\vartheta +2)}}{\left({\left(z_2-z_1\right)}^{\rho }{\displaystyle \frac{1}{2^{\rho }}}\right)}^j{\left({\displaystyle \frac{1}{\rho j+\vartheta +2}}\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ & \times {\left({\displaystyle \frac{1}{\rho j+\vartheta +2}}\right)}^{{\displaystyle \frac{1}{q}}}[{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{(\rho j+\vartheta +4)|f^{″}\left(z_1\right){|}^q+(\rho j+\vartheta +2){|f^{″}\left(z_2\right)|}^q}{2(\rho j+\vartheta +3)}}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ & +{\left(|f^{″}\left({\nu }_1\right){|}^q+{|f^{″}\left({\nu }_2\right)|}^q-{\displaystyle \frac{(\rho j+\vartheta +2)|f^{″}\left(z_1\right){|}^q+(\rho j+\vartheta +4){|f^{″}\left(z_2\right)|}^q}{2(\rho j+\vartheta +3)}}\right)}^{{\displaystyle \frac{1}{q}}}].\hfill \end{array}$$

which completes the demonstration. □

Remark 14.

If we take $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1$, in Theorem 6, then Theorem 6 gives [28] [Theorem 4].

Remark 15.

If we consider $q=1,$ in Theorem 6, then Theorem 6 reduces in Theorem 4.

4. Applications

The applications of the obtained results in the area of numerical analysis and special functions are demonstrated in this section.

4.1. Numerical Trapezoidal and Midpoint Quadrature Formula

For $0<{\nu }_1<{\nu }_2,$ let $P:{\nu }_1={\mathcal{X}}_0<{\mathcal{X}}_1<\cdots <{\mathcal{X}}_{n-1}<{\mathcal{X}}_n={\nu }_2$ be a partition of $[{\nu }_1,{\nu }_2],$ and $x_{k,1},x_{k,2}\in [{\mathcal{X}}_k,{\mathcal{X}}_{k+1}]$ for all $k=0,1,2,\dots ,n-1$. Let consider the following quadrature formulas:

$$\begin{array}{ccc}\hfill {\int }_{{\nu }_1+{\nu }_2-z_2}^{{\nu }_1+{\nu }_2-z_1}f\left(u\right)du=T(P,f)+E(P,f),& \hfill & \hfill {\int }_{{\nu }_1+{\nu }_2-z_2}^{{\nu }_1+{\nu }_2-z_1}f\left(u\right)du=M(P,f)+E^{\ast }(P,f),\end{array}$$

where

$$T(P,f)=\sum _{k=o}^{n-1}\left({\displaystyle \frac{f({\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,2})+f({\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,1})}{2}}\right)h_k,$$

and

$$M(P,f)=\sum _{k=0}^{n-1}f\left({\mathcal{X}}_k+{\mathcal{X}}_{k+1}-{\displaystyle \frac{x_{k,1}+x_{k,2}}{2}}\right)h_k,$$

are the trapezoidal version and Midpoint version, respectively, and $E(P,f),$$E^{\ast }(P,f)$ are denote their associated approximation errors and $h_k=x_{k,2}-x_{k,1}.$

Using above notations, we are in position to prove the following error estimations.

Proposition 1.

Under the assumption of Theorem 2, if we take $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1,$ then the following inequality holds:

$$\left|E(P,f)\right|\le {\displaystyle \frac{1}{12}}\sum _{k=0}^{n-1}{h}_k^3{\left[|f^{″}\left({\mathcal{X}}_k\right){|}^q+{|f^{″}\left({\mathcal{X}}_{k+1}\right)|}^q-{\displaystyle \frac{|f^{″}\left(x_{k,1}\right){|}^q+{|f^{″}\left(x_{k,2}\right)|}^q}{2}}\right]}^{{\displaystyle \frac{1}{q}}}.$$

Proof. 

Using the Theorem 2 on sub-interval $[{\mathcal{X}}_k,{\mathcal{X}}_{k+1}]$ of closed interval $[{\nu }_1,{\nu }_2]$ and choosing $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1,$ for all $k=0,1,2,\dots ,n-1,$ we have

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{f({\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,2})+f({\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,1})}{2}}(x_{k,2}-x_{k,1})-{\int }_{{\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,2}}^{{\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,1}}f\left(u\right)du|\hfill \\ & \le {\displaystyle \frac{{(x_{k,2}-x_{k,1})}^3}{12}}{\left[|f^{″}\left({\mathcal{X}}_k\right){|}^q+{|f^{″}\left({\mathcal{X}}_{k+1}\right)|}^q-{\displaystyle \frac{|f^{″}\left(x_{k,1}\right){|}^q+{|f^{″}\left(x_{k,2}\right)|}^q}{2}}\right]}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Summing inequality above over k from 0 to $n-1$, we achieve the desired inequality. □

A parallel application of Theorem 4 gives the result for the midpoint formula.

Proposition 2.

Under the assumption of Theorem 4, if we take $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1,$ then the following inequality holds the following:

$$\left|E^{\ast }(P,f)\right|\le {\displaystyle \frac{1}{24}}\sum _{k=0}^{n-1}{h}_k^3\left[|f^{″}\left({\mathcal{X}}_k\right)|+|f^{″}\left({\mathcal{X}}_{k+1}\right)|-{\displaystyle \frac{|f^{″}\left(x_{k,1}\right)|+|f^{″}\left(x_{k,2}\right)|}{2}}\right].$$

Proof. 

Using the Theorem 4 on sub-interval $[{\mathcal{X}}_k,{\mathcal{X}}_{k+1}]$ of closed interval $[{\nu }_1,{\nu }_2]$ and choosing $\sigma \left(0\right)=1,$$j=0,$ and $\vartheta =1,$ for all $k=0,1,2,\dots ,n-1,$ we have

$$\begin{array}{cc}\hfill & |{\int }_{{\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,2}}^{{\mathcal{X}}_k+{\mathcal{X}}_{k+1}-x_{k,1}}f\left(u\right)du-f\left({\mathcal{X}}_k+{\mathcal{X}}_{k+1}-{\displaystyle \frac{x_{k,1}+x_{k,2}}{2}}\right)(x_{k,2}-x_{k,1})|\hfill \\ & \le {\displaystyle \frac{{(x_{k,2}-x_{k,1})}^3}{24}}\left[|f^{″}\left({\mathcal{X}}_k\right)|+|f^{″}\left({\mathcal{X}}_{k+1}\right)|-{\displaystyle \frac{|f^{″}\left(x_{k,1}\right)|+|f^{″}\left(x_{k,2}\right)|}{2}}\right].\hfill \end{array}$$

Summing inequality above over k from 0 to $n-1$ and using the properties of the modulus, we achieve the desired inequality. □

Remark 16.

We can obtain, in addition, more inequalities for the quadrature formulas by utilizing the same approach like in Proposition 1, and Proposition 2. We do not write here their proofs, and the details will be for the interested reader.

Next, we address several new applications of our previous results for the modified Bessel functions and also for q-digamma function.

4.2. Modified Bessel Function

We consider the function ${\Im }_{\theta }:R\to (0,1]$ given as follows:

$${\Im }_{\theta }\left(\tau \right)=2^{\theta }\Gamma (1+\theta ){\tau }^{-s}{I}_{\theta }\left(\tau \right).$$

For this, we recall the representation of modified Bessel functions, which is presented, in [43]:

$${\Im }_{\theta }\left(\tau \right)=\sum _{u=0}^{\infty }{\displaystyle \frac{{\left(\frac{\tau }{2}\right)}^{\theta +2u}}{u!\Gamma (\theta +u+1)}},$$

The first and n th-order derivative formula’s ${\Im }_{\theta }\left(\tau \right)$, are presented in [20]:

$${\Im }_{\theta }^{\prime }\left(\tau \right)={\displaystyle \frac{\tau }{2(1+\theta )}}{\Im }_{\theta +1}\left(\tau \right),$$

$$\begin{array}{c}\hfill {\displaystyle \frac{{\partial }^n{\Im }_{\theta }\left(\tau \right)}{\partial {\tau }^n}}=2^{n-2\theta }\sqrt{\pi }{\tau }^{\theta -n}\Gamma (1+\theta ){}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{1+\theta -n}{2}},{\displaystyle \frac{2+\theta -n}{2}},1+\theta ;{\displaystyle \frac{{\tau }^2}{4}}\right),\end{array}$$

where ${}_{2}F_3(.,.;.,.,.;.)$ is the hypergeometric function given in [20]:

$$\begin{array}{c}\hfill {}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{1+\theta -n}{2}},{\displaystyle \frac{2+\theta -n}{2}},1+\theta ;{\displaystyle \frac{{\tau }^2}{4}}\right)=\sum _{r=0}^{\infty }{\displaystyle \frac{{\left(\frac{1+\theta }{2}\right)}_r{\left(\frac{2+\theta }{2}\right)}_r{\tau }^{2r}}{{\left(\frac{1+\theta -n}{2}\right)}_r{\left(\frac{2+\theta -n}{2}\right)}_r{(1+\theta )}_r{4}^{r}r!}}.\end{array}$$

Proposition 3.

Let $0<{\nu }_1<{\nu }_2$ and $\theta \ge 1.$ Then, for all $z_1,z_2\in [{\nu }_1,{\nu }_2]$, real numbers, we get

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{({\nu }_1+{\nu }_2-z_2){\Im }_{\theta +1}({\nu }_1+{\nu }_2-z_2)+({\nu }_1+{\nu }_2-z_1){\Im }_{\theta +1}({\nu }_1+{\nu }_2-z_1)}{4(1+\theta )}}\hfill \\ & -{\displaystyle \frac{1}{z_2-z_1}}\left[{\Im }_{\theta }({\nu }_1+{\nu }_2-z_1)-{\Im }_{\theta }({\nu }_1+{\nu }_2-z_2)\right]|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{3.2^{2\theta }}}\sqrt{\pi }\Gamma (1+\theta )[2{\nu }_1^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{{\nu }_1^2}{4}}\right)\hfill \\ & +2{\nu }_2^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{{\nu }_2^2}{4}}\right)\hfill \\ & -z_1^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{z_1^2}{4}}\right)\hfill \\ & -z_2^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{z_2^2}{4}}\right)].\hfill \end{array}$$

Proof. 

The demonstration is derived directly by Theorem 2, taking into account that $f\left(\eta \right)={\Im }_{\theta }^{\prime }\left(\eta \right),$$\vartheta =q=1,$$j=0$ and that $\sigma \left(0\right)=1$. □

Proposition 4.

Let $0<{\nu }_1<{\nu }_2$ and $\theta \ge 1.$ Then, for all $z_1,z_2\in [{\nu }_1,{\nu }_2]$ that are real numbers, we have the following:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{1}{z_2-z_1}}\left[{\Im }_{\theta }({\nu }_1+{\nu }_2-z_1)-{\Im }_{\theta }({\nu }_1+{\nu }_2-z_2)\right]-{\displaystyle \frac{2({\nu }_1+{\nu }_2)-z_1-z_2}{4(1+\theta )}}{\Im }_{\theta +1}\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{3.2^{2\theta +1}}}\sqrt{\pi }\Gamma (1+\theta )[2{\nu }_1^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{{\nu }_1^2}{4}}\right)\hfill \\ & +2{\nu }_2^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{{\nu }_2^2}{4}}\right)\hfill \\ & -z_1^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{z_1^2}{4}}\right)\hfill \\ & -z_2^{\theta -3}{}_{2}F_3\left({\displaystyle \frac{1+\theta }{2}},{\displaystyle \frac{2+\theta }{2}};{\displaystyle \frac{\theta -2}{2}},{\displaystyle \frac{\theta -1}{2}},1+\theta ;{\displaystyle \frac{z_2^2}{4}}\right)].\hfill \end{array}$$

Proof. 

The proof results directly from Theorem 4, taking into account that $f\left(\eta \right)={\Im }_{\theta }^{\prime }\left(\eta \right),$$\vartheta =1,$$j=0$ and that $\sigma \left(0\right)=1$. □

Remark 17.

We can obtain more inequalities for the modified Bessel function of the first kind by using the same approach as in Propositions 3 and 4. We will omit their demonstrations, and the details will be left to the interested reader.

4.3. q-Digamma Function

We revisit the necessary notion, of the q-digamma function and also its mathematical representations; assume that $0<q<1.$ The q-digamma function ${\mathcal{X}}_q\left(u\right)$, [11,44], is given as follows:

$$\begin{array}{cc}\hfill {\mathcal{X}}_q\left(u\right)& =-ln(1-q)+ln\left(q\right)\sum _{i=0}^{\infty }{\displaystyle \frac{q^{i+u}}{1-q^{i+u}}}\hfill \\ & =-ln(1-q)+ln\left(q\right)\sum _{i=1}^{\infty }{\displaystyle \frac{q^{iu}}{1-q^i}}.\hfill \end{array}$$

If $1<q$ and $u>1,$ the q-digamma function ${\mathcal{X}}_q\left(u\right)$ is represented as follows:

$$\begin{array}{cc}\hfill {\mathcal{X}}_q\left(u\right)& =-ln(q-1)+ln\left(q\right)[u-{\displaystyle \frac{1}{2}}-\sum _{i=0}^{\infty }{\displaystyle \frac{q^{-(i+u)}}{1-q^{-(i+u)}}}]\hfill \\ & =-ln(q-1)+ln\left(q\right)[u-{\displaystyle \frac{1}{2}}-\sum _{i=1}^{\infty }{\displaystyle \frac{q^{-iu}}{1-q^{-i}}}].\hfill \end{array}$$

The above notion tell us that for $q>0,$ the function ${\mathcal{X}}_q^{\prime }\left(u\right)$ is completely monotonic on $(0,\infty )$. This thing implies that it is a convex mapping. Therefore, we can formulate the next important findings about the function q-digamma.

Proposition 5.

By Theorem 2, we have the following:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{{\mathcal{X}}_q^{\prime }({\nu }_1+{\nu }_2-z_2)+{\mathcal{X}}_q^{\prime }({\nu }_1+{\nu }_2-z_1)}{2}}-{\displaystyle \frac{{\mathcal{X}}_q({\nu }_1+{\nu }_2-z_1)-{\mathcal{X}}_q({\nu }_1+{\nu }_2-z_2)}{z_2-z_1}}|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{12}}{\left(|{\mathcal{X}}_q^{‴}\left({\nu }_1\right){|}^q+{|{\mathcal{X}}_q^{‴}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|{\mathcal{X}}_q^{‴}\left(z_1\right){|}^q+{|{\mathcal{X}}_q^{‴}\left(z_2\right)|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Proof. 

The demonstration is derived directly by Theorem 2, if we take $f\left(\eta \right)={\mathcal{X}}_q^{\prime }\left(\eta \right),$$\vartheta =1,$$j=0$ and that $\sigma \left(0\right)=1$. □

Proposition 6.

From Theorem 4, we get

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{{\mathcal{X}}_q({\nu }_1+{\nu }_2-z_1)-{\mathcal{X}}_q({\nu }_1+{\nu }_2-z_2)}{z_2-z_1}}-{\mathcal{X}}_q^{\prime }\left({\nu }_1+{\nu }_2-{\displaystyle \frac{z_1+z_2}{2}}\right)|\hfill \\ & \le {\displaystyle \frac{{(z_2-z_1)}^2}{24}}{\left(|{\mathcal{X}}_q^{‴}\left({\nu }_1\right){|}^q+{|{\mathcal{X}}_q^{‴}\left({\nu }_2\right)|}^q-{\displaystyle \frac{|{\mathcal{X}}_q^{‴}\left(z_1\right){|}^q+3{|{\mathcal{X}}_q^{‴}\left(z_2\right)|}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}}.\hfill \end{array}$$

Proof. 

The demonstration is derived directly from Theorem 4, if we take $f\left(\eta \right)={\mathcal{X}}_q^{\prime }\left(\eta \right),$$\vartheta =1,$$j=0$ and using that $\sigma \left(0\right)=1$. □

Remark 18.

By applying our main results, we may derive a number of attractive inequalities for the q-digamma function, by using the same method like in relations previously proved. We do not include their demonstrations.

5. Visual Interpretation and Computational Evaluation

The key findings of this work are presented here through visual interpretation and computational Evaluation, providing valuable context for understanding the theoretical results.

Example 1.

We take $f:[{\nu }_1,{\nu }_2]\to R$, $f\left(x\right)={(x+a_1)}^5$, ${\nu }_1=l, {\nu }_2=4l$, $z_1=2l,$$z_2=3l$. In addition, $\sigma \left(0\right)=1,$ and $w=0$. We can see that the hypothesis of Theorem 2, are valid if $q\ge 1$ and $l, a_1>0$ so the corresponding inequality takes place and we can write the following:

$$|{\displaystyle \frac{{(2l+a_1)}^5+{(3l+a_1)}^5}{2}}-{\displaystyle \frac{{\Gamma }_{(}\vartheta +1)}{2l^{\vartheta }\Gamma \left(\vartheta \right)}}[{\int }_{2l}^{3l}{(3l-\eta )}^{\vartheta -1}{(\eta +a_1)}^{5}d\eta +{\int }_{2l}^{3l}{(\eta -2l)}^{\vartheta -1}{(\eta +a_1)}^{5}d\eta ]|$$

$$\le {\displaystyle \frac{l^2\vartheta }{2}}{\displaystyle \frac{\Gamma (\vartheta +1)}{\Gamma (\vartheta +3)}}{(20{(l+a_1)}^{3q}+20{(4l+a_1)}^{3q}-{\displaystyle \frac{20{(2l+a_1)}^{3q}+20{(3l+a_1)}^{3q}}{2}})}^{{\displaystyle \frac{1}{q}}}$$

If we denote by C the integral, ${\int }_{2l}^{3l}{(3l-\eta )}^{\vartheta -1}{(\eta +a_1)}^{5}d\eta$ and by D the integral ${\int }_{2l}^{3l}{(\eta -2l)}^{\vartheta -1}{(\eta +a_1)}^{5}d\eta$, then the left member becomes the following:

$$M_s=|{\displaystyle \frac{{(2l+a_1)}^5+{(3l+a_1)}^5}{2}}-{\displaystyle \frac{\vartheta }{2l^{\vartheta }}}(C+D)|.$$

The right member will be

$$M_d={\displaystyle \frac{10l^2\vartheta }{(\vartheta +1)(\vartheta +2)}}{\left({(l+a_1)}^{3q}+{(4l+a_1)}^{3q}-{\displaystyle \frac{{(2l+a_1)}^{3q}+{(3l+a_1)}^{3q}}{2}}\right)}^{{\displaystyle \frac{1}{q}}}$$

Changing the variable, after calculus, we get

$$\begin{array}{cc}\hfill C=l^{\vartheta }[& {\displaystyle \frac{{(3l+a_1)}^5}{\vartheta }}-5{\displaystyle \frac{l}{\vartheta +1}}{(3l+a_1)}^4+10{\displaystyle \frac{l^2}{\vartheta +2}}{(3l+a_1)}^3-10{\displaystyle \frac{l^3}{\vartheta +3}}{(3l+a_1)}^2\hfill \\ & +5{\displaystyle \frac{l^4}{\vartheta +4}}(3l+a_1)-{\displaystyle \frac{l^5}{\vartheta +5}}],\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill D=l^{\vartheta }[& {\displaystyle \frac{{(2l+a_1)}^5}{\vartheta }}+5{\displaystyle \frac{l}{\vartheta +1}}{(2l+a_1)}^4+10{\displaystyle \frac{l^2}{\vartheta +2}}{(2l+a_1)}^3+10{\displaystyle \frac{l^3}{\vartheta +3}}{(2l+a_1)}^2\hfill \\ & +5{\displaystyle \frac{l^4}{\vartheta +4}}(2l+a_1)+{\displaystyle \frac{l^5}{\vartheta +5}}],\hfill \end{array}$$

respectively.

Therefore, our inequality, by calculus, becomes the following:

$$|{\displaystyle \frac{5}{2(\vartheta +1)}}({(2l+a_1)}^4-{(3l+a_1)}^4)+5{\displaystyle \frac{l}{\vartheta +2}}({(2l+a_1)}^3+{(3l+a_1)}^3)$$

$$+5{\displaystyle \frac{l^2}{\vartheta +3}}{(2l+a_1)}^2-{(3l+a_1)}^2+{\displaystyle \frac{5l^3}{2(\vartheta +4)}}(5l+2a_1)|\le$$

$$\le {\displaystyle \frac{10l\vartheta }{(\vartheta +1)(\vartheta +2)}}{\left({(l+a_1)}^{3q}+{(4l+a_1)}^{3q}-{\displaystyle \frac{{(2l+a_1)}^{3q}+{(3l+a_1)}^{3q}}{2}}\right)}^{{\displaystyle \frac{1}{q}}}.$$

(8)

(a) If we consider the particular case when, $l=1$ and $q=1$ then we have

$$|{\displaystyle \frac{5}{2(\vartheta +1)}}({(2+a_1)}^4-{(3+a_1)}^4)+{\displaystyle \frac{5}{\vartheta +2}}({(2+a_1)}^3+{(3+a_1)}^3)$$

$$+{\displaystyle \frac{5}{\vartheta +3}}{(2+a_1)}^2-{(3+a_1)}^2+{\displaystyle \frac{5}{2(\vartheta +4)}}(5+2a_1)| \le$$

$$\le {\displaystyle \frac{10\vartheta }{(\vartheta +1)(\vartheta +2)}}\left({(1+a_1)}^3+{(4+a_1)}^3-{\displaystyle \frac{{(2+a_1)}^3+{(3+a_1)}^3}{2}}\right).$$

(b) If we consider the particular case when, $\vartheta =1$ and $q=1$ then we obtain

$$|{\displaystyle \frac{5}{4}}({(2l+a_1)}^4-{(3l+a_1)}^4)+5{\displaystyle \frac{l}{3}}({(2l+a_1)}^3+{(3l+a_1)}^3)$$

$$+5{\displaystyle \frac{l^2}{4}}{(2l+a_1)}^2-{(3l+a_1)}^2+{\displaystyle \frac{l^3}{2}}(5l+2a_1)| \le$$

$$\le {\displaystyle \frac{5l}{3}}\left({(l+a_1)}^3+{(4l+a_1)}^3-{\displaystyle \frac{{(2l+a_1)}^3+{(3l+a_1)}^3}{2}}\right).$$

(c) If we consider the particular case when, $a_1=0$ then by calculus, we get

$$| -{\displaystyle \frac{13}{2(\vartheta +1)}}+{\displaystyle \frac{7}{\vartheta +2}}-{\displaystyle \frac{1}{\vartheta +3}}+{\displaystyle \frac{1}{2(\vartheta +4)}}|$$

$$\le {\displaystyle \frac{2}{5(\vartheta +1)(\vartheta +2)}}{\left(1+{64}^q-{\displaystyle \frac{8^q+{27}^q}{2}}\right)}^{{\displaystyle \frac{1}{q}}}$$

when $q\ge 1.$

In Figure 1a, the two surfaces represent the left and the right member of the inequality from Theorem 2, graphically illustrated in the particular case of previous example, when $l=1,q=1$ and $\vartheta ,a_1\in [0.01,2.6]$ are variables. The left member $M_s$ is the blue surface and the right member $M_d$ is the magenta surface. In Figure 1b, the two lines represent the left member (blue line) and the right member (magenta line) of the same inequality, graphically illustrated for previous example, when $l=1,q=1$, $a_1=2$, and $\vartheta \in [0.01,2.6]$ are variables.

In Figure 2a, the two members of inequality from Theorem 2 are represented by surfaces in blue and magenta, respectively, but when $\vartheta =1,q=1$ and $l, a_1\in [0.01, 2.6]$ are variables. The same members are represented in Figure 2b, but in this case when $\vartheta =1,q=1$, $a_1=1$ and $l\in [0.01,2.6]$ we obtain a two dimensional graphic.

In addition, Figure 3a shows the same members of the same inequality from Figure 2a, but when $\vartheta =1,q=1$, $l=1$ and $a_1\in [0.01, 2.6]$. In Figure 3b, the same left and right members are represented by surfaces in blue and magenta, respectively, but when $a_1=0$ and $\vartheta , q\in [1, 2.6]$ are variables.

From previous examples we can notice that in inequality (8) we have four variables, $\vartheta , a_1, l$ and $q.$ Therefore we will generate 10 random values in interval $[1, 2.6]$ for each variable by using a Matlab R2023b software version command and then compute the left-member and the right-member of the inequality (8) from Example 1 also by a Matlab in

Table 1. Random verification of inequality (8) from Theorem 2.

	$\mathit{\vartheta }$	${\mathit{a}}_1$	l	q	${\mathit{M}}_{\mathit{s}}$	${\mathit{M}}_{\mathit{d}}$
i = 1	1.5627	1.1214	1.2595	1.7209	108.8025	287.4075
i = 2	2.3293	1.0863	2.2709	1.1341	495.7516	1277.7
i = 3	1.9364	1.8493	1.4979	1.3664	229.3053	506.7888
i = 4	1.8796	2.2467	1.8457	2.4613	539.3167	1402.9
i = 5	2.4675	2.4944	1.2650	1.2438	148.9598	268.0879
i = 6	1.4573	1.2078	1.9632	2.3213	536.6874	1642.3
i = 7	2.2115	1.9101	1.4208	1.8613	172.9154	412.3582
i = 8	2.2060	1.7510	2.0465	2.5938	498.1211	1435.7
i = 9	1.6087	1.0190	2.1027	1.1251	561.3629	1437.5
i = 10	1.9085	1.5394	2.1970	1.7083	681.8574	1908.2

from below for validating our results.

It can be seen that the right member, $M_d$ is biger than the left member $M_s$ so the inequality (8) from Theorem 2 is verified.

Example 2. 

Assume that all the properties of Theorem 4 are fulfilled, and considering the mapping $f\left(\eta \right)={\displaystyle \frac{m^2}{(r+3m)(r+2m)}}{\eta }^{{\displaystyle \frac{r}{m}}+3}$ defined on $R^{+}$ with $r\ge 1$ and $m>1$ be convex functions and $f^{″}\left(\eta \right)={\eta }^{{\displaystyle \frac{r}{m}}+1}$ with $r\ge 1$ and $m>1$ be also convex mapping. Fixing the following values ${\nu }_1=1,$${\nu }_2=6,$$z_1=2,$$z_2=4,$$\vartheta =1,$$j=0$ and the fact that $\sigma \left(0\right)=1,$ then

$$\begin{array}{c}\hfill M_{s1}=\left|{\displaystyle \frac{m^2}{(r+3m)(r+2m)}}\left[{\displaystyle \frac{m}{(r+4m)}}\left({\displaystyle \frac{5^{\frac{r}{m}+4}-3^{\frac{r}{m}+4}}{2}}\right)-4^{{\displaystyle \frac{r}{m}}+3}\right]\right|,\end{array}$$

and

$$\begin{array}{c}\hfill M_{d1}={\displaystyle \frac{1}{6}}\left[1+6^{{\displaystyle \frac{r}{m}}+1}-2^{{\displaystyle \frac{r}{m}}}\left(1+2^{{\displaystyle \frac{r}{m}}+1}\right)\right].\end{array}$$

• In Figure 4a, we used the particular case from Example 2 when ${\nu }_1=1, {\nu }_2=6, z_1=2, z_2=4, \vartheta =1, j=0, \sigma \left(0\right)=1$ and $r,m\in [1,6]$ are variables. The parameters r and m are varied over the interval $[1,6]$ to compare the left member (blue) and right member (magenta) of Theorem 4.
• In Figure 4b, the left side (blue) and right side (magenta) of Theorem 4 are compared by setting $m=3$ and varying r over the interval $[1,6].$
• In Figure 4c, the left side (blue) and right sides (magenta) of Theorem 4 are compared by setting $r=4$ and varying m over the interval $[1,5].$

Figure 4. For function f in Example 2 and parameters ${\nu }_1=1,{\nu }_2=6,$$z_1=2,z_2=4$, $\vartheta =1,$$j=0, \sigma \left(0\right)=1$, the graphics in 3D and 2D, respectively, of the left members (blue) and right members (magenta) of the inequality from Example 2, Theorem 4 are given when (a) $r,m\in [1,6]$; (b) $r\in [1,6]$ and $m=3$; (c) $m\in [1,5]$ and $r=4.$

Figure 4. For function f in Example 2 and parameters ${\nu }_1=1,{\nu }_2=6,$$z_1=2,z_2=4$, $\vartheta =1,$$j=0, \sigma \left(0\right)=1$, the graphics in 3D and 2D, respectively, of the left members (blue) and right members (magenta) of the inequality from Example 2, Theorem 4 are given when (a) $r,m\in [1,6]$; (b) $r\in [1,6]$ and $m=3$; (c) $m\in [1,5]$ and $r=4.$

We will generate now 10 random values in interval $[1.5, 6]$ for each variable by using a Matlab R2023b software version command and then compute the left member and the right member of the inequality from Theorem 4 from Example 2 by using Matlab software version R2023b in

Table 2. Random verification of inequality from Theorem 4.

	r	m	${\mathit{M}}_{\mathit{s}1}$	${\mathit{M}}_{\mathit{d}1}$
i = 1	5.1663	2.2093	17.4693	56.8169
i = 2	5.5761	5.8677	2.5036	4.0889
i = 3	2.0714	5.8073	1.0948	1.3015
i = 4	5.6102	3.6824	5.5705	12.2446
i = 5	4.3456	5.1013	2.1823	3.3814
i = 6	1.9389	2.1385	2.3557	3.7589
i = 7	2.7532	3.3979	2.0593	3.1203
i = 8	3.9610	5.6208	1.1775	2.5444
i = 9	5.8088	5.0649	3.2939	5.9693
i = 10	5.8420	5.8177	2.6990	4.5363

from below.

It can be seen that the right member, $M_{d1}$ is larger than the left member $M_{s1}$ so the inequality of Theorem 4 is verified.

Example 3. 

Assume that all the properties of Theorem 5 are fulfilled, and considering the mapping $f\left(\eta \right)={\displaystyle \frac{m^2}{(r+3m)(r+2m)}}{\eta }^{{\displaystyle \frac{r}{m}}+3}$ defined on $R^{+}$ with $r\ge 1$ and $m>1$ be convex functions and $f^{″}\left(\eta \right)={\eta }^{{\displaystyle \frac{r}{m}}+1}$ with $r\ge 1$ and $m>1$ be also convex mapping. Fixing the following values ${\nu }_1=1,$${\nu }_2=4,$$z_1=2,$$z_2=3,$$\vartheta =1,$$j=0$ and the fact that $\sigma \left(0\right)=1,$ then

$$\begin{array}{c}\hfill M_{s2}=\left|{\displaystyle \frac{m^2}{(r+3m)(r+2m)}}\left[{\displaystyle \frac{m}{(r+4m)}}\left(3^{{\displaystyle \frac{r}{m}}+4}-2^{{\displaystyle \frac{r}{m}}+4}\right)-{\left({\displaystyle \frac{5}{2}}\right)}^{{\displaystyle \frac{r}{m}}+3}\right]\right|,\end{array}$$

and

$$\begin{array}{c}\hfill M_{d2}={\displaystyle \frac{1}{16\sqrt[p]{2p+1}}}\left[{\left(1+4^{{\displaystyle \frac{q(r+m)}{m}}}-{\displaystyle \frac{3\times 2^{\frac{q(r+m)}{m}}+3^{\frac{q(r+m)}{m}}}{4}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left(1+4^{{\displaystyle \frac{q(r+m)}{m}}}-{\displaystyle \frac{2^{\frac{q(r+m)}{m}}+3^{\frac{q(r+m)+m}{m}}}{4}}\right)}^{{\displaystyle \frac{1}{q}}}\right].\end{array}$$

• In Figure 5a, it is represented for parameters ${\nu }_1=1, {\nu }_2=6, z_1=2, z_2=4, \vartheta =1, j=0, \sigma \left(0\right)=1$ and the function f from Example 3 the graphic for the left member and the graphic for the right member of the inequality from Example 3 in 3D when $r,m\in [2, 6]$. The parameters r and m are varied over the interval $[1, 6]$ to compare the left member (blue) and right member (magenta) of Theorem 5.
• In Figure 5b, the left side (blue) and right side (magenta) of Theorem 5 are compared by setting $m=4$ and varying r over the interval $[1, 6].$
• In Figure 5c, the left side (blue) and right sides (magenta) of Theorem 5 from Example 3 are compared by setting $r=3$ and varying m over the interval $[2, 5].$

Figure 5. For function f in Example 3 and parameters ${\nu }_1=1,{\nu }_2=6,$$z_1=2,z_2=4$, $\vartheta =1,$$j=0, \sigma \left(0\right)=1$, the graphics in 3D and 2D of the left members (blue) and right members (magenta) of the inequality from Example 3, see Theorem 5 are given when (a) $r,m\in [2,6]$. (b) $r\in \left[1.6\right]$ and $m=4$; (c) $m\in [2,5]$ and $r=3.$

Figure 5. For function f in Example 3 and parameters ${\nu }_1=1,{\nu }_2=6,$$z_1=2,z_2=4$, $\vartheta =1,$$j=0, \sigma \left(0\right)=1$, the graphics in 3D and 2D of the left members (blue) and right members (magenta) of the inequality from Example 3, see Theorem 5 are given when (a) $r,m\in [2,6]$. (b) $r\in \left[1.6\right]$ and $m=4$; (c) $m\in [2,5]$ and $r=3.$

We will generate now 10 random values in interval $[1.5, 6]$ by using a Matlab command and then compute the left member and the right member of the inequality from Example 3 by using Matlab R2023b software version in

Table 3. Random verification of inequality in Example 3, Theorem 5.

	r	m	q	p	${\mathit{M}}_{\mathit{s}2}$	${\mathit{M}}_{\mathit{d}2}$
i = 1	3.4743	2.7421	4.8807	1.2577u	0.3345	1.0612
i = 2	3.2170	4.5587	2.6479	1.6068	0.1993	0.5093
i = 3	4.9448	4.4479	3.7768	1.3601	0.2898	0.8761
i = 4	5.0784	2.2318	4.6458	1.2743	0.8505	4.3323
i = 5	2.3409	2.0355	5.5091	1.2218	0.3003	0.8923
i = 6	3.7039	3.7426	5.8168	1.2076	0.2590	0.7108
i = 7	3.5051	5.8188	3.9625	1.3376	0.1812	0.4256
i = 8	4.4084	3.0317	2.1238	1.8898	0.3976	1.5402
i = 9	4.6921	4.1337	2.1718	1.8534	0.2962	0.9697
i = 10	4.8961	2.5072	2.6588	1.6029	0.6307	2.9945

from below in order to illustrate the validity of our results.

It can be seen that the right member, $M_{d2}$ is biger than the left member $M_{s2}$ so the inequality in Example 3 Theorem 5 is verified.

Example 4.

Assume that all the properties of Theorem 6 are fulfilled, and considering the mapping $f\left(\eta \right)={\displaystyle \frac{m^2}{(r+3m)(r+2m)}}{\eta }^{{\displaystyle \frac{r}{m}}+3}$ defined on $R^{+}$ with $r\ge 1$ and $m>1$ be convex functions and $f^{″}\left(\eta \right)={\eta }^{{\displaystyle \frac{r}{m}}+1}$ with $r\ge 1$ and $m>1$ be also convex mapping. Fixing the following values ${\nu }_1=0,$${\nu }_2=1,$$z_1={\displaystyle \frac{1}{3}},$$z_2={\displaystyle \frac{2}{3}},$$\vartheta =1,$$j=0$ and the fact that $\sigma \left(0\right)=1,$ then

$$\begin{array}{c}\hfill M_{s3}=\left|{\displaystyle \frac{m^2}{(r+3m)(r+2m)}}\left[{\displaystyle \frac{3m}{(r+4m)}}\left({\left({\displaystyle \frac{2}{3}}\right)}^{{\displaystyle \frac{r}{m}}+4}-{\left({\displaystyle \frac{1}{3}}\right)}^{{\displaystyle \frac{r}{m}}+4}\right)-{\left({\displaystyle \frac{1}{2}}\right)}^{{\displaystyle \frac{r}{m}}+3}\right]\right|,\end{array}$$

and

$$\begin{array}{c}\hfill M_{d3}={\displaystyle \frac{1}{432}}\left[{\left(1-{\displaystyle \frac{5{\left(\frac{1}{3}\right)}^{\frac{q(r+m)}{m}}+3{\left(\frac{2}{3}\right)}^{\frac{q(r+m)}{m}}}{8}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left(1-{\displaystyle \frac{3{\left(\frac{1}{3}\right)}^{\frac{q(r+m)}{m}}+5{\left(\frac{2}{3}\right)}^{\frac{q(r+m)}{m}}}{8}}\right)}^{{\displaystyle \frac{1}{q}}}\right].\end{array}$$

Figure 6a–c shows the graphics for the left member and right member of the inequality from Theorem 6 from Example 4 in 3D in the case when the parameters are $j=0,\vartheta =1,{\nu }_1=0,{\nu }_2=1$, $z_1={\displaystyle \frac{1}{3}},z_2={\displaystyle \frac{2}{3}},\sigma \left(0\right)=1,$ and the function f is as in Example 4.

• In Figure 6a, the variables are considered as being r, and m when $r,m\in [2, 6]$ and in Theorem 6, $q={\displaystyle \frac{3}{2}}>1.$
• In Figure 6b, the variables are considered as being m and q and r is fixed, $r=4$. We considered here $m,q\in [2, 6]$.
• In Figure 6c, we fixed m, $m=3$ and the variables will be r and q, when $r,q\in [2, 6].$

Like in Example 3, it can be seen that in all the three figures the magenta surface is above the blue surface so the inequality from Theorem 6 is verified.

The graphical representations were done by using the Matlab software version R2023b. Also in all figures, it can be seen that the right member which correspond to magenta line/surface is above the the left member which correspond to blue line/surface so the inequalities from Examples 2–4 are fulfilled.

Remark 19.  (a) In the case from Figure 6a, if we take $r\in [2, 6]$ as variable and we particularize m then in 2D we can plot a graph between the left and the right-hand side of Theorem 6 too. (b) In the case from Figure 6a, if we take $m\in [2, 5]$ as a variable and we particularize r then in 2D we can plot a graph between the left and the right-hand side of Theorem 6. (c) Similar cases with 2D graphics can be obtained also from Figure 6b,c.

It can be seen that in inequality from Theorem 6, if we consider the three variables as in Example 4, r, m and q and if and we generate 10 random values in interval $[2, 6]$, for each variable by using a Matlab command, and then compute the left member and the right member of previous mentioned inequality by using Matlab software, we will obtain

Table 4. Random verification of inequality from Example 4, Theorem 6.

	r	m	q	${\mathit{M}}_{\mathit{s}3}$	${\mathit{M}}_{\mathit{d}3}$
i = 1	3.6690682	5.1210082	2.9391196	0.0014183	0.0045221
i = 2	2.1986177	3.5589553	3.4126342	0.0015168	0.0045541
i = 3	5.6108644	2.9667651	5.2847761	0.0006429	0.0046287
i = 4	5.7791487	3.6156485	2.0616137	0.0007820	0.0044964
i = 5	3.9634563	2.3858181	2.1720952	0.0007498	0.0045240
i = 6	3.9570105	2.5278931	2.6759601	0.0007996	0.0045751
i = 7	3.3508776	5.7682023	4.5964618	0.0015554	0.0046027
i = 8	5.6002153	5.8245381	4.9268895	0.0012011	0.0046202
i = 9	3.4769871	4.3008343	4.5909838	0.0013324	0.0046120
i = 10	2.4448110	2.2391181	3.8036948	0.0010997	0.0046051

.

It can be seen that the right member, $M_{d3}$ is larger than the left member $M_{s3}$ so the inequality from Theorem 6, Example 4 is verified.

6. Conclusions

This paper begins with the development of two novel identity for the Raina’s fractional integral operator. Based on these identities, we establish several integral inequalities associated with the left-hand side and right-hand side of Hermite–Hadamard-type inequalities in the framework of the Raina’s fractional integral operator. Furthermore, we show that the proposed results refine and generalize certain previously established results in the field of integral inequalities. Next, we provide applications to error estimates for trapezoidal and midpoint quadrature formulas and to analytical evaluations involving modified Bessel functions of the first kind and q-digamma functions, and we show the validity of the proposed inequalities in numerical integration and analysis of special functions. Finally, to enhance the understanding and clarity of the newly derived inequalities, we provide several examples along with their graphical and numerical illustrations. In comparison to classical calculus, our results are more advantageous as they highlight the specific case of previously established bounds when $\vartheta =1,$$j=0$ and $\sigma \left(0\right)=1$. We hope that our approach and results will inspire readers to explore this topic further.

Future studies may explore similar inequalities for different fractional integrals, new Hermite–Hadamard–Mercer-type inequalities could be derived. Moreover, our results can be extended in the future through establishing more inequalities for different kinds of convexity, particularly in the fields of q-calculus, stochastic processes, and interval calculus. Numerous areas of the pure and practical sciences, such as statistical analysis, optimization problems, and differential equations, might be helpful significantly across the tools and methods presented in this study.