Hybrid Integral Inequalities on Fractal Set

Abstract

In this study, we introduce a new hybrid identity that effectively combines Newton–Cotes and Gauss quadrature, allowing us to recover well-known formulas such as Simpson’s second rule and the left- and right-Radau two-point rules, among others. Building upon this flexible framework, we establish several new biparametrized fractal integral inequalities for functions whose local fractional derivatives are of a generalized convex type. In addition to employing tools from local fractional calculus, our approach utilizes the Hölder inequality, the power mean inequality, and a refined version of the latter. Further results are also derived using the concept of generalized concavity. To support our theoretical findings, we provide a graphical example that illustrates the validity of the obtained results, along with some practical applications that demonstrate their effectiveness.

1. Introduction

A function $\xi$ is recognized as convex on the interval I if for any $u_1,u_2$ in I and t in $[0,1]$, the following condition holds [1]:

$$\xi (t{u}_1+(1-t)u_2)\le t\xi \left(u_1\right)+(1-t)\xi \left(u_2\right)$$

The significance of convexity becomes even more pronounced when considering integral inequalities, a fundamental area of study in mathematical analysis. Integral inequalities serve as powerful tools for estimating and bounding the values of integrals, playing a pivotal role in the study of differential equations and the stability of solutions. For some papers dealing with inequalities via convexity, we refer the readers to [2,3,4].

The authors in [4] introduced the subsequent $3/8$-Simpson-type inequality for convex first-order derivatives:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left(\xi \left(\phi \right)+3\xi \left(\frac{2\phi +\psi }{3}\right)+3\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{\phi }{\stackrel{\psi }{\int }}\xi \left(u\right)du\right|\hfill \\ \hfill \le & \frac{25\left(\psi -\phi \right)}{576}\left(\left|{\xi }^{\prime }\left(\phi \right)\right|+\left|{\xi }^{\prime }\left(\psi \right)\right|\right).\hfill \end{array}$$

In [5], Laribi et al. offered an enhancement of the previously mentioned outcome for the same class of functions in the following manner:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left(\xi \left(\phi \right)+3\xi \left(\frac{2\phi +\psi }{3}\right)+3\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{\phi }{\stackrel{\psi }{\int }}\xi \left(u\right)du\right|\hfill \\ \hfill \le & \frac{\psi -\phi }{9}\left(\frac{157}{1536}\left(\left|{\xi }^{\prime }\left(\phi \right)\right|+\left|{\xi }^{\prime }\left(\psi \right)\right|\right)+\frac{443}{1536}\left(\left|{\xi }^{\prime }\left(\frac{2\phi +\psi }{3}\right)\right|+\left|{\xi }^{\prime }\left(\frac{\phi +2\psi }{3}\right)\right|\right)\right).\hfill \end{array}$$

In [6], Meftah et al. derived the following two-point left-Radau-type inequality for differentiable convex functions:

$$\left|\frac{1}{4}\left(\xi \left(\phi \right)+3\xi \left(\frac{\phi +2\psi }{3}\right)\right)-\frac{1}{\psi -\phi }\underset{\phi }{\stackrel{\psi }{\int }}\xi \left(u\right)du\right|\le \frac{\left(\psi -\phi \right)}{9}\left(\frac{157\left|{\xi }^{\prime }\left(\phi \right)\right|+379\left|{\xi }^{\prime }\left(\frac{\phi +2\psi }{3}\right)\right|+64\left|{\xi }^{\prime }\left(\psi \right)\right|}{192}\right).$$

In the same context, Rebiai et al. [7] provided the two-point right-Radau-type inequality for differentiable convex functions in the following manner:

$$\left|\frac{1}{4}\left(3\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{\phi }{\stackrel{\psi }{\int }}\xi \left(u\right)du\right|\le \frac{\psi -\phi }{9}\left(\frac{1}{3}\left|\xi \left(\phi \right)\right|+\frac{379}{192}\left|\xi \left(\frac{2\phi +\psi }{3}\right)\right|+\frac{157}{192}\left|\xi \left(\psi \right)\right|\right).$$

For more results on Radau-type inequalities, we refer the readers to [8,9].

The enigmatic world of fractal sets, characterized by their self-similar structures and intricate geometries, continues to fascinate and challenge researchers. These sets, which defy traditional Euclidean geometry, necessitate the use of advanced mathematical tools for their analysis and comprehension. Among these tools, local fractional calculus stands out as a cornerstone for investigating fractal sets. This calculus extends the concept of classical differentiation and integration to non-integer orders, allowing for a more nuanced exploration of the irregular and fragmented dimensions intrinsic to fractal structures [10,11,12].

Local fractional calculus not only provides a deeper understanding of fractal sets but also bridges the gap between classical mathematical concepts and the complex nature of fractals. It is within this framework that the notion of convexity gains new dimensions. Traditionally, convexity has been a central concept in mathematical analysis and optimization, offering a structured and robust approach to understanding the behavior of functions and sets. In the realm of fractal sets, convexity interacts uniquely with the fractional dimensions, leading to novel insights and results; see [13,14].

In [15], Mo et al. expanded the concept of convexity to the framework of fractal sets in the following manner.

A function $\xi :I\subseteq \mathbb{R}\to {\mathbb{R}}^{\gamma }$ is recognized as generalized convex on I if for any $u_1,u_2\in I$ and $t\in [0,1]$, the following inequality holds:

$$\xi \left(t{u}_1+\left(1-t\right)u_2\right)\le t^{\gamma }\xi \left(u_1\right)+{\left(1-t\right)}^{\gamma }\xi \left(u_2\right).$$

After introducing such a concept, numerous researchers devoted their efforts to investigating local fractional integrals and establishing error bounds for various quadrature rules. In [16], Khan et al. established generalized trapezium-type inequalities using generalized convexity. Saleh et al. provided a companion of Ostrowski’s inequality for the same class of functions in [17]. Sarikaya et al. presented Simpson-type inequalities in [18].

In [19], Luo and coauthors derived Fejér–Hermite–Hadamard-type inequalities involving generalized h-convexity, while Sun established Ostrowski’s inequality for the same class of functions in [20]. Du et al. provided certain integral inequalities considering generalized m-convexity on fractal sets in [21] and Hermite–Hadamard- and Pachpatte-type inequalities for generalized subadditive functions in the fractal sense in [22]. Zhang and coauthors explored properties and weighted parametric inequalities via generalized $(m,h)$-preinvex mappings in [23]. Yu et al. proposed Bullen-type inequalities as well as certain error bounds for parameterized integral inequalities in [24,25], respectively. In [26], Sun obtained Hermite–Hadamard-type local fractional integral inequalities with a Mittag–Leffler kernel for generalized preinvex functions. In [27], Razzaq et al. established Hermite–Hadamard inequalities for generalized $(m-F)$-convex functions, while Napoles Valdes investigated Milne’s inequality via generalized modified $(h,m)$-convex functions in [28]. For further studies via different types of generalized convexity, we refer the reader to [29,30,31,32].

Beyond the aforementioned works, further interesting parametric studies were carried out by Du and Yan, and Xu et al. in [33,34], respectively. More importantly, Li et al. conducted a multiparameter analysis leading to a wide range of results based on generalized $(s,P)$-convexity in [35], while Xu et al. also performed a multiparametrized investigation of local fractional integral inequalities via generalized $\alpha$-convexity in [36]. For additional studies in this direction, we refer the reader to [37,38,39,40].

In Ref. [41], the authors established the following Simpson second formula inequality related to local fractional integrals:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & {\left(\psi -\phi \right)}^{\gamma }\left({\left(\frac{528}{13824}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+{\left(\frac{1008}{13824}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left(\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right),\hfill \end{array}$$

where $\left|{\xi }^{\left(\gamma \right)}\right|$ is a generalized convex function.

In [42], Bin-Mohsin et al. provided an extension of the two-point left-Radau inequality to fractal sets in the following manner:

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\left({\left(\frac{31}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{7}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|\right.\hfill \\ \hfill & +\left({\left(\frac{43}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-{\left(\frac{27}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|\hfill \\ \hfill & +\left(2^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+{\left(\frac{1}{4}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|\hfill \\ \hfill & +\left.\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right),\hfill \end{array}$$

where $\left|{\xi }^{\left(\gamma \right)}\right|$ is a generalized convex function.

In this study, we delve into the intersection of these profound concepts. By introducing a novel hybrid identity that amalgamates the strengths of Newton–Cotes and Gauss quadrature, we recover renowned formulas and extend our understanding to encompass the left- and right-Radau two-point rules, among others. Our exploration of biparametrized fractal integral inequalities for functions with generalized convex local fractional derivatives paves the way for innovative theoretical developments and practical applications, demonstrating the immense potential and versatility of our results.

2. Preliminaries

In this section, we offer the fundamental definitions, propositions, and lemmas essential for grasping the concept of local calculus as introduced by Yang [13,43]. For $0<\gamma \le 1$, let ${\mathbb{R}}^{\gamma }$ denote the $\gamma$-type fractal set of real line numbers.

If $u_1^{\gamma },u_2^{\gamma },$ and $u_3^{\gamma }$ belong to ${\mathbb{R}}^{\gamma }$, then the following properties are true:

•

$u_1^{\gamma }+u_2^{\gamma }$ and $u_1^{\gamma }{u}_2^{\gamma }$ belong to ${\mathbb{R}}^{\gamma }$.

•

$u_1^{\gamma }+u_2^{\gamma }=u_2^{\gamma }+u_1^{\gamma }={(u_1+u_2)}^{\gamma }={(u_2+u_1)}^{\gamma }$.

•

$u_1^{\gamma }+(u_2^{\gamma }+u_3^{\gamma })={(u_1+u_2)}^{\gamma }+u_3^{\gamma }$.

•

$u_1^{\gamma }{u}_2^{\gamma }=u_2^{\gamma }{u}_1^{\nu }={\left(u_1{u}_2\right)}^{\gamma }={\left(u_2{u}_1\right)}^{\gamma }$.

•

$u_1^{\gamma }\left(u_2^{\gamma }{u}_3^{\gamma }\right)=\left(u_1^{\gamma }{u}_2^{\gamma }\right)u_3^{\gamma }$.

•

$u_1^{\gamma }(u_2^{\gamma }+u_3^{\gamma })=u_1^{\gamma }{u}_2^{\gamma }+u_1^{\gamma }{u}_3^{\gamma }$.

•

$u_1^{\gamma }+0^{\gamma }=0^{\gamma }+u_1^{\gamma }=u_1^{\gamma }$ and $u_1^{\gamma }{1}^{\gamma }=1^{\gamma }{u}_1^{\gamma }=u_1^{\gamma }$.

Definition 1

([43]). A function $\xi :\mathbb{R}\to {\mathbb{R}}^{\gamma }$ is local fractional continuous at $u_0$ if

$$\forall ϵ>0,\exists \delta >0:\left|\xi \left(u\right)-\xi \left(u_0\right)\right|<{ϵ}^{\gamma }$$

holds for $|u-u_0|<\delta$, where $ϵ,\delta \in \mathbb{R}$. We denote the set of all local fractional continuous functions on $\left(\phi ,\psi \right)$ by $C_{\gamma }\left(\phi ,\psi \right)$.

Definition 2

([43]). The local fractional derivative of $\xi \left(u\right)$ of order γ at $u=u_0$ is defined as

$${\xi }^{\left(\gamma \right)}\left(u_0\right)={\left.\frac{d^{\gamma }\xi \left(u\right)}{d{u}^{\gamma }}\right|}_{u=u_0}=\underset{u\to u_0}{lim}\frac{{\Delta }^{\gamma }\left(\xi \left(u\right)-\xi \left(u_0\right)\right)}{{\left(u-u_0\right)}^{\gamma }},$$

where ${\Delta }^{\gamma }\left(\xi \left(u\right)-\xi \left(u_0\right)\right)\cong \mathrm{\Gamma }\left(\gamma +1\right)\left(\xi \left(u\right)-\xi \left(u_0\right)\right)$.

If there exists ${\xi }^{(k+1)\gamma }\left(u\right)=\stackrel{(k+1)\mathrm{times}}{\overbrace{D^{\gamma }{D}^{\gamma }\dots D^{\gamma }}}\xi \left(u\right)$ for any $u\in I\subseteq R$, then we say that $\xi \in D_{(k+1)\gamma }\left(I\right)$, where $k=0,1,2,3,\dots$

Definition 3

([43]). Let $\xi \left(x\right)\in C_{\gamma }\left[\phi ,\psi \right]$. Then, the local fractional integral is defined by

$${}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)=\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{\phi }{\stackrel{\psi }{\int }}\xi \left(t\right){\left(dt\right)}^{\gamma }=\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{\Delta t\to 0}{lim}\stackrel{N-1}{\sum _{j=0}}\xi \left(t_j\right){\left(\Delta t_j\right)}^{\gamma }$$

with $\Delta t_j=t_{j+1}-t_j$ and $\Delta t=max\left\{\Delta t_1,\Delta t_2,\dots ,\Delta t_{N-1}\right\}$, where $\left[t_j,t_{j+1}\right],j=0,1,\dots ,N-1$, and $\phi =t_0<t_1<\dots <t_N=\psi$ is a partition of interval $\left[\phi ,\psi \right]$.

Here, it follows that ${}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)=0$ if $\phi =\psi$ and ${}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)={-}_{\psi }{I}_{\phi }^{\gamma }\xi \left(u\right)$ if $\phi <\psi$. If for any $u\in [\phi ,\psi ]$ there exists ${}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)$, then we denote $\xi \left(u\right)\in I_u^{\gamma }\left[\phi ,\psi \right]$.

Lemma 1

([43]).

1.

(Local fractional integration is anti-differentiation) Suppose that $\xi \left(u\right)={\zeta }^{\left(\gamma \right)}\left(u\right)\in C_{\gamma }\left[\phi ,\psi \right]$. Then, we have

$${}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)=\zeta \left(\psi \right)-\zeta \left(\phi \right).$$

2.

(Local fractional integration by parts) Suppose that $\xi ,\zeta \in D_{\gamma }\left[\phi ,\psi \right]$ and ${\xi }^{\left(\gamma \right)}\left(u\right),{\zeta }^{\left(\gamma \right)}\left(u\right)\in C_{\gamma }\left[\phi ,\psi \right]$.Then, we have

$${}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right){\zeta }^{\left(\gamma \right)}\left(u\right)={\left.\xi \left(u\right)\zeta \left(u\right)\right|}_{\phi }^{\psi }-{}_{\phi }{I}_{\psi }^{\gamma }{\xi }^{\left(\gamma \right)}\left(u\right)\zeta \left(u\right).$$

Lemma 2

([43]). For $s\in \mathbb{R}$, we have the following identities:

$$\begin{array}{cc}\hfill \frac{d^{\gamma }{u}^{s\gamma }}{d{u}^{\gamma }}=& \frac{\mathrm{\Gamma }\left(1+s\gamma \right)}{\mathrm{\Gamma }\left(1+\left(s-1\right)\gamma \right)}{u}^{\left(s-1\right)\gamma },\hfill \\ \hfill \frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}{u}^{s\gamma }{\left(du\right)}^{\gamma }=& \frac{\mathrm{\Gamma }\left(1+s\gamma \right)}{\mathrm{\Gamma }\left(1+\left(s+1\right)\gamma \right)}\left({\psi }^{\left(s+1\right)\gamma }-{\phi }^{\left(s+1\right)\gamma }\right).\hfill \end{array}$$

Lemma 3

(Generalized Hölder’s inequality [44]). Let $\xi ,\zeta \in C_{\gamma }\left[\phi ,\psi \right]$ and ${\left|\xi \left(u\right)\right|}^p,{\left|\zeta \left(u\right)\right|}^q$, where $p,q>1$ with ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$ are both integrable under the frame of the fractal spaces. Then, we have

$$\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}\left|\xi \left(u\right)\zeta \left(u\right)\right|{\left(du\right)}^{\gamma }\le {\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}{\left|\xi \left(u\right)\right|}^p{\left(du\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}{\left|\zeta \left(u\right)\right|}^q{\left(du\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}.$$

Lemma 4

(Generalized power mean inequality [44]). Let $\xi ,\zeta \in C_{\gamma }\left[\phi ,\psi \right]$ and $\left|\xi \left(u\right)\right|$, $\left|\xi \left(u\right)\right|{\left|\zeta \left(u\right)\right|}^q$, where $q>1$ is integrable under the frame of the fractal spaces. Then, we have

$$\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}\left|\xi \left(u\right)\zeta \left(u\right)\right|{\left(du\right)}^{\gamma }\le {\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}\left|\xi \left(u\right)\right|{\left(du\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}\left|\xi \left(u\right)\right|{\left|\zeta \left(u\right)\right|}^q{\left(du\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}.$$

Lemma 5

(Improved generalized power mean inequality [45]). Let $\xi ,\zeta \in C_{\gamma }\left[\phi ,\psi \right]$ and $\left|\xi \left(u\right)\right|,\left|\xi \left(u\right)\right|{\left|\zeta \left(u\right)\right|}^q$, where $q>1$ is integrable under the frame of the fractal spaces. Then, we have

$$\begin{array}{cc}\hfill & \frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}\left|\xi \left(u\right)\zeta \left(u\right)\right|{\left(du\right)}^{\gamma }\hfill \\ \hfill \le & {\left(\frac{1}{\psi -\phi }\right)}^{\gamma }\left({\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}{\left(\psi -u\right)}^{\gamma }\left|\xi \left(u\right)\right|{\left(du\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\right.{\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}{\left(\psi -u\right)}^{\gamma }\left|\xi \left(u\right)\right|{\left|\zeta \left(u\right)\right|}^q{\left(du\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}{\left(u-\phi \right)}^{\gamma }\left|\xi \left(u\right)\right|{\left(du\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\left.{\left(\frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}\underset{\phi }{\stackrel{\psi }{\int }}{\left(u-\phi \right)}^{\gamma }\left|\xi \left(u\right)\right|{\left|\zeta \left(u\right)\right|}^q{\left(du\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

3. Main Results

In this section, we will present some results that we will use later.

Lemma 6.

Let $\xi :I\to {\mathbb{R}}^{\gamma }$ be a local fractional differentiable on $I^{\circ }$, $\phi ,\psi \in I^{\circ }$ with $\phi <\psi$, and ${\xi }^{\left(\gamma \right)}\in C_{\gamma }\left[\phi ,\psi \right]$. Then, for all real numbers $\gamma >0$ and $\lambda ,\vartheta \ge 0$ with $\lambda +\vartheta \ne 0$, the following equality holds:

$$\begin{array}{cc}\hfill & \frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\hfill \\ \hfill =& \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right){\left(dt\right)}^{\gamma }\right.\hfill \\ \hfill & -\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right)}^{\gamma }{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right){\left(dt\right)}^{\gamma }\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right){\left(dt\right)}^{\gamma }\right),\hfill \end{array}$$

where

$${}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)=\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{\phi }{\stackrel{{\displaystyle \frac{2\phi +\psi }{3}}}{\int }}\xi \left(u\right){\left(du\right)}^{\gamma }+\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{{\displaystyle \frac{2\phi +\psi }{3}}}{\stackrel{{\displaystyle \frac{\phi +2\psi }{3}}}{\int }}\xi \left(u\right){\left(du\right)}^{\gamma }+\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{{\displaystyle \frac{\phi +2\psi }{3}}}{\stackrel{\psi }{\int }}\xi \left(u\right){\left(du\right)}^{\gamma }.$$

Proof. 

Let

$$\mathcal{I}={\mathcal{I}}_1-{\mathcal{I}}_2+{\mathcal{I}}_3,$$

(1)

where

$$\begin{array}{cc}\hfill {\mathcal{I}}_1=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right){\left(dt\right)}^{\gamma },\hfill \\ \hfill {\mathcal{I}}_2=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right)}^{\gamma }{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right){\left(dt\right)}^{\gamma }\hfill \end{array}$$

and

$${\mathcal{I}}_3=\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right){\left(dt\right)}^{\gamma }.$$

Using Lemma 1, ${\mathcal{I}}_1$ gives

$$\begin{array}{cc}\hfill & {\mathcal{I}}_1={\left.\frac{3^{\gamma }}{{\left(\psi -\phi \right)}^{\gamma }}{\left(t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\xi \left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right)\right|}_0^1\hfill \\ \hfill & -\frac{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}\underset{0}{\stackrel{1}{\int }}\xi \left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right){\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{3^{\gamma }{\left(\lambda +4\vartheta \right)}^{\gamma }}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }{\left(\psi -\phi \right)}^{\gamma }}\xi \left(\frac{2\phi +\psi }{3}\right)+\frac{9^{\gamma }{\lambda }^{\gamma }}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }{\left(\psi -\phi \right)}^{\gamma }}\xi \left(\phi \right)\hfill \\ \hfill & -\frac{9^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{2\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\underset{\phi }{\stackrel{{\displaystyle \frac{2\phi +\psi }{3}}}{\int }}\xi \left(u\right){\left(du\right)}^{\gamma }.\hfill \end{array}$$

(2)

Similarly, we obtain

$$\begin{array}{cc}\hfill & {\mathcal{I}}_2={\left.\frac{3^{\gamma }}{{\left(\psi -\phi \right)}^{\gamma }}{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right)}^{\gamma }\xi \left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right)\right|}_0^1\hfill \\ \hfill & +\frac{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}\underset{0}{\stackrel{1}{\int }}\xi \left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right){\left(dt\right)}^{\gamma }\hfill \\ \hfill =& -\frac{3^{\gamma }{\left(5\lambda -\vartheta \right)}^{\gamma }}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }{\left(\psi -\phi \right)}^{\gamma }}\xi \left(\frac{\phi +2\psi }{3}\right)-\frac{3^{\gamma }{\left(5\vartheta -\lambda \right)}^{\gamma }}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }{\left(\psi -\phi \right)}^{\gamma }}\xi \left(\frac{2\phi +\psi }{3}\right)\hfill \\ \hfill & +\frac{9^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{2\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\underset{{\displaystyle \frac{2\phi +\psi }{3}}}{\stackrel{{\displaystyle \frac{\phi +2\psi }{3}}}{\int }}\xi \left(u\right){\left(du\right)}^{\gamma }\hfill \end{array}$$

(3)

and

$$\begin{array}{cc}\hfill {\mathcal{I}}_3=& {\left.\frac{3^{\gamma }}{{\left(\psi -\phi \right)}^{\gamma }}{\left(t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\xi \left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right)\right|}_0^1\hfill \\ \hfill & -\frac{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}\underset{0}{\stackrel{1}{\int }}\xi \left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right){\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{9^{\gamma }{\vartheta }^{\gamma }}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }{\left(\psi -\phi \right)}^{\gamma }}\xi \left(\psi \right)+\frac{3^{\gamma }{\left(4\lambda +\vartheta \right)}^{\gamma }}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }{\left(\psi -\phi \right)}^{\gamma }}\xi \left(\frac{\phi +2\psi }{3}\right)\hfill \\ \hfill & -\frac{9^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{2\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\underset{{\displaystyle \frac{\phi +2\psi }{3}}}{\stackrel{\psi }{\int }}\xi \left(u\right){\left(du\right)}^{\gamma }.\hfill \end{array}$$

(4)

Using (2)–(4) in (1), and then multiplying the resulting equality by $\frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}$, we obtain the desired result. □

Theorem 1.

Let $\xi :\left[\phi ,\psi \right]\to {\mathbb{R}}^{\gamma }$ be a local fractional differentiable on $\left[\phi ,\psi \right]$ such that $\xi \in D_{\gamma }\left[\phi ,\psi \right]$ and ${\xi }^{\left(\gamma \right)}\in C_{\gamma }\left[\phi ,\psi \right]$ with $\phi <\psi$. If $\left|{\xi }^{\left(\gamma \right)}\right|$ is generalized convex on $\left[\phi ,\psi \right]$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\mathcal{L}}_1\left(\gamma ,\lambda ,\vartheta \right)\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left({\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right)\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|\right.\hfill \\ \hfill & +\left.\left({\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right)\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|+{\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right)\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right),\hfill \end{array}$$

where ${\mathcal{L}}_i$ for $i=1,2,\dots ,6$ are defined as in (5)–(10).

Proof. 

From Lemma 6 and the properties of the modulus, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right)\right|{\left(dt\right)}^{\gamma }\right.\hfill \\ \hfill & +\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right)\right|{\left(dt\right)}^{\gamma }\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right)\right|{\left(dt\right)}^{\gamma }\right).\hfill \end{array}$$

Using the generalized convexity of $\left|{\xi }^{\left(\gamma \right)}\right|$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+t^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|\right){\left(dt\right)}^{\gamma }\right.\hfill \\ \hfill & +\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|+t^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|\right){\left(dt\right)}^{\gamma }\hfill \\ \hfill & +\left.\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|+t^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right){\left(dt\right)}^{\gamma }\right)\hfill \\ \hfill =& \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\mathcal{L}}_1\left(\gamma ,\lambda ,\vartheta \right)\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left({\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right)\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|\right.\hfill \\ \hfill & +\left.\left({\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right)\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|+{\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right)\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right),\hfill \end{array}$$

where we used

$$\begin{array}{cc}\hfill {\mathcal{L}}_1\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(1-t\right)}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+2^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{t}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+2^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma },\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(1-t\right)}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ 2^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\hfill \\ +\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill {\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{t}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ 2^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\hfill \\ +\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ {\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(1-t\right)}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+2^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill {\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{t}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+2^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }.\hfill \end{array}$$

(10)

Thus, the proof is complete. □

Corollary 1.

In Theorem 1, using the generalized convexity of $\left|{\xi }^{\left(\gamma \right)}\right|$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\frac{3^{\gamma }{\mathcal{L}}_1\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|\right.\hfill \\ \hfill & +\left.\frac{{\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right)+3^{\gamma }{\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right).\hfill \end{array}$$

Remark 1.

1.

If we attempt to take $\vartheta =0$, Theorem 1 will be reduced to Theorem 2 from [42].

2.

If we attempt to take $\vartheta =0$, Corollary 1 will be reduced to Corollary 2 from [42].

Corollary 2.

In Theorem 1, taking $\lambda =0$, we obtain the right-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|\right.\hfill \\ \hfill & +\left(2^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+{\left(\frac{1}{4}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|\hfill \\ \hfill & +\left({\left(\frac{43}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-{\left(\frac{27}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|\hfill \\ \hfill & +\left.\left({\left(\frac{31}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{7}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right).\hfill \end{array}$$

Corollary 3.

In Corollary 2, if $\gamma \to 1$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4}\left(3\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{0}{\stackrel{1}{\int }}\xi \left(u\right)du\right|\hfill \\ \hfill \le & \frac{\psi -\phi }{9}\left(\frac{32\left|{\xi }^{\prime }\left(\phi \right)\right|+152\left|{\xi }^{\prime }\left(\frac{2\phi +\psi }{3}\right)\right|+75\left|{\xi }^{\prime }\left(\frac{\phi +2\psi }{3}\right)\right|+83\left|{\xi }^{\prime }\left(\psi \right)\right|}{192}\right).\hfill \end{array}$$

Corollary 4.

In Theorem 1, taking $\lambda =\vartheta$, we obtain Simpson’s second formula inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\left({\left(\frac{131}{256}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{35}{256}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left(\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right)\right.\hfill \\ \hfill & +\left.\left({\left(\frac{421}{256}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{133}{256}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left(\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|\right)\right).\hfill \end{array}$$

Corollary 5.

In Corollary 4, if $\gamma \to 1$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left(\xi \left(\phi \right)+3\xi \left(\frac{2\phi +\psi }{3}\right)+3\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{0}{\stackrel{1}{\int }}\xi \left(u\right)du\right|\hfill \\ \hfill \le & \frac{\psi -\phi }{9}\left(\frac{157}{1536}\left(\left|{\xi }^{\prime }\left(\phi \right)\right|+\left|{\xi }^{\prime }\left(\psi \right)\right|\right)+\frac{443}{1536}\left(\left|{\xi }^{\prime }\left(\frac{2\phi +\psi }{3}\right)\right|+\left|{\xi }^{\prime }\left(\frac{\phi +2\psi }{3}\right)\right|\right)\right),\hfill \end{array}$$

which is the same result given in Corollary 2.1 from [5].

Corollary 6.

In Corollary 1, taking $\lambda =0$, we obtain the right-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{{27}^{\gamma }}\left(\left({\left(\frac{155}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+{\left(\frac{5}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|\right.\hfill \\ \hfill & +\left.\left({\left(\frac{73}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+{\left(\frac{103}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right).\hfill \end{array}$$

Corollary 7.

In Corollary 1, taking $\vartheta =\lambda$, we obtain Simpson’s second formula:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{552}{256}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{168}{256}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left(\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|\right).\hfill \end{array}$$

Corollary 8.

In Corollary 7, if $\gamma \to 1$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left(\xi \left(\phi \right)+3\xi \left(\frac{2\phi +\psi }{3}\right)+3\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{0}{\stackrel{1}{\int }}\xi \left(u\right)du\right|\hfill \\ \hfill \le & \frac{25\left(\psi -\phi \right)}{576}\left(\left|{\xi }^{\prime }\left(\phi \right)\right|+\left|{\xi }^{\prime }\left(\psi \right)\right|\right),\hfill \end{array}$$

which is the same result given in Remark 3 from [4].

Theorem 2.

Assume that all the assumptions of Theorem 1 are satisfied. If ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$ is generalized convex, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left(\frac{{\left(3\lambda \right)}^{p+1}+{\left(\lambda +4\vartheta \right)}^{p+1}}{{\left(4\left(\lambda +\vartheta \right)\right)}^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\mathrm{\Gamma }\left(1+p\gamma \right)}{\rm Y}\left(\gamma ,\lambda ,\vartheta \right)\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{{\left(4\lambda +\vartheta \right)}^{p+1}+{\left(3\vartheta \right)}^{p+1}}{{\left(4\left(\lambda +\vartheta \right)\right)}^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

where Υ is defined by (12) and $p,q>1$ with ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$.

Proof. 

From Lemma 6 and the properties of the modulus among with a generalized Hölder inequality, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\right.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\left.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Using the generalized convexity of ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\right.\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \times \left.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill =& \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left({\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\mathrm{\Gamma }\left(1+p\gamma \right)}{\rm Y}\left(\gamma ,\lambda ,\vartheta \right)\right)}^{{\displaystyle \frac{1}{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left({\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

where we have used

$$\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }=\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\left({\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{\gamma },$$

(11)

$$\begin{array}{cc}\hfill {\rm Y}\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\left({\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}-{\left(\frac{\lambda -5\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{\gamma }\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ \frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\left({\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{\gamma }\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ \frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\left({\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}-{\left(\frac{\vartheta -5\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{\gamma }\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(12)

$$\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }=\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\left({\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{\gamma }$$

(13)

and

$$\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left(dt\right)}^{\gamma }=\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left(dt\right)}^{\gamma }=\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}.$$

The proof is complete. □

Corollary 9.

In Theorem 2, using the generalized convexity of ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{2^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left({\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{5^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q}{6^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\mathrm{\Gamma }\left(1+p\gamma \right)}{\rm Y}\left(\gamma ,\lambda ,\vartheta \right)\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q}{2^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left({\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+2^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q}{3^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 10.

In Theorem 2, taking $\vartheta =0$, we obtain the left-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left(\frac{1+3^{p+1}}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{5^{p+1}-1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 11.

In Theorem 2, taking $\lambda =0$, we obtain the right-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left({\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{5^{p+1}-1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{1+3^{p+1}}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 12.

In Theorem 2, taking $\lambda =\vartheta$, we obtain Simpson’s second formula inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left(\frac{3^{p+1}+5^{p+1}}{8^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{1}{2}\right)}^{\gamma }{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{3^{p+1}+5^{p+1}}{8^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 13.

In Corollary 12, if $\gamma \to 1$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left(\xi \left(\phi \right)+3\xi \left(\frac{2\phi +\psi }{3}\right)+3\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{0}{\stackrel{1}{\int }}\xi \left(u\right)du\right|\hfill \\ \hfill \le & \frac{\psi -\phi }{9}{\left(\frac{1}{p+1}\right)}^{{\displaystyle \frac{1}{p}}}\left({\left(\frac{3^{p+1}+5^{p+1}}{8^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{{\left|{\xi }^{\prime }\left(\phi \right)\right|}^q+{\left|{\xi }^{\prime }\left(\frac{2\phi +\psi }{3}\right)\right|}^q}{2}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.\frac{1}{2}{\left(\frac{{\left|{\xi }^{\prime }\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\prime }\left(\frac{\phi +2\psi }{3}\right)\right|}^q}{2}\right)}^{{\displaystyle \frac{1}{q}}}+{\left(\frac{3^{p+1}+5^{p+1}}{8^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{{\left|{\xi }^{\prime }\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\prime }\left(\psi \right)\right|}^q}{2}\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

which is the same result given in Corollary 3.5 from [2].

Corollary 14.

In Corollary 9, taking $\vartheta =0$, we obtain the left-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{2^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\left({\left(\frac{3^{p+1}+1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{5^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}{6^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left(\frac{5^{p+1}-1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}{2^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}+{\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+2^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q}{3^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 15.

In Corollary 9, taking $\lambda =0$, we obtain the right-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{2^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\left({\left(\frac{5^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}{6^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left(\frac{5^{p+1}-1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}{2^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}+{\left(\frac{3^{p+1}+1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}{\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+2^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q}{3^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 16.

In Corollary 9, taking $\vartheta =\lambda$, we obtain Simpson’s second formula inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{2^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}\left({\left(\frac{1}{2}\right)}^{\gamma }{\left(\frac{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}{2^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +\left.{\left(\frac{3^{p+1}+5^{p+1}}{8^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left({\left(\frac{5^{\gamma }\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}{6^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}+{\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+2^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q}{3^{\gamma }}\right)}^{{\displaystyle \frac{1}{q}}}\right)\right).\hfill \end{array}$$

Theorem 3.

Assume that all the assumptions of Theorem 1 are satisfied. If ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$ is generalized convex for $q>1$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\frac{9{\lambda }^2+{\left(\lambda +4\vartheta \right)}^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma \left(1-{\displaystyle \frac{1}{q}}\right)}\right.\hfill \\ \hfill & \times {\left({\mathcal{L}}_1\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+\gamma \right)}\Delta \left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{9{\vartheta }^2+{\left(4\lambda +\vartheta \right)}^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma \left(1-{\displaystyle \frac{1}{q}}\right)}{\left({\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+{\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

where Δ is defined by (14) and ${\mathcal{L}}_i$ (for $i=1,\dots ,6$) are defined as (5)–(10), respectively.

Proof. 

From Lemma 6 and the properties of the modulus with a generalized power mean inequality, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Using the generalized convexity of ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill =& \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}{\left(\frac{9{\lambda }^2+{\left(\lambda +4\vartheta \right)}^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \times {\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(1-t\right)}^{\gamma }{\left(dt\right)}^{\gamma }+\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{t}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\Delta \left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(1-t\right)}^{\gamma }{\left(dt\right)}^{\gamma }+\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{t}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}{\left(\frac{{\left(4\lambda +\vartheta \right)}^2+9{\vartheta }^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left.{\left(\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(1-t\right)}^{\gamma }{\left(dt\right)}^{\gamma }+\frac{{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{t}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill =& \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\frac{9{\lambda }^2+{\left(\lambda +4\vartheta \right)}^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma \left(1-{\displaystyle \frac{1}{q}}\right)}\right.\hfill \\ \hfill & \times {\left({\mathcal{L}}_1\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+\gamma \right)}\Delta \left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left({\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{{\left(4\lambda +\vartheta \right)}^2+9{\vartheta }^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma \left(1-{\displaystyle \frac{1}{q}}\right)}\hfill \\ \hfill & \times \left.{\left({\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+{\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

where we have used (5)–(10) and

$$\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }=\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}{\left(\frac{9{\lambda }^2+{\left(\lambda +4\vartheta \right)}^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma },$$

$$\begin{array}{cc}\hfill \Delta \left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}{\left({\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^2-{\left(\frac{\lambda -5\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^2\right)}^{\gamma }\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ \frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}{\left({\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^2+{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^2\right)}^{\gamma }\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ \frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}{\left({\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^2-{\left(\frac{\vartheta -5\lambda }{4\left(\lambda +\vartheta \right)}\right)}^2\right)}^{\gamma }\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(14)

and

$$\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }=\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}{\left(\frac{{\left(4\lambda +\vartheta \right)}^2+9{\vartheta }^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma }.$$

This completes the proof. □

Corollary 17.

In Theorem 3, using the generalized convexity of ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left({\left(\frac{9{\lambda }^2+{\left(\lambda +4\vartheta \right)}^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma \left(1-{\displaystyle \frac{1}{q}}\right)}\right.{\left(\frac{3^{\gamma }{\mathcal{L}}_1\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{{\mathcal{L}}_2\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+\gamma \right)}\Delta \left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{2^{\gamma }{\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{{\mathcal{L}}_3\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{L}}_4\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{{\left(4\lambda +\vartheta \right)}^2+9{\vartheta }^2}{16{\left(\lambda +\vartheta \right)}^2}\right)}^{\gamma \left(1-{\displaystyle \frac{1}{q}}\right)}\left.{\left(\frac{{\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{2^{\gamma }{\mathcal{L}}_5\left(\gamma ,\lambda ,\vartheta \right)+3^{\gamma }{\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 18.

In Theorem 3, taking $\vartheta =0$, we obtain the left-Radau type-inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\frac{5^{\gamma }}{8^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}\left(\left(\frac{{31}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{7^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q\right.\right.\hfill \\ \hfill & +{\left.\left(\frac{3^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{5^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{3^{\gamma }}{2^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\left(\frac{5^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 19.

In Theorem 3, taking $\lambda =0$, we obtain the right-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{3^{\gamma }}{2^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+\left(\frac{5^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{5^{\gamma }}{8^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\left(\frac{3^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{5^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+\left(\frac{{31}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{7\mathrm{\Gamma }\left(1+\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 20.

In Theorem 3, taking $\lambda =\vartheta$, we obtain Simpson’s second formula inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\frac{{17}^{\gamma }}{{32}^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}\right.\left(\left(\frac{{131}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{35}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q\right.\hfill \\ \hfill & {\left.+\left(\frac{{229}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{69}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{2^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{3^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{{17}^{\gamma }}{{32}^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}\left(\left(\frac{{229}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{69}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right.\hfill \\ \hfill & \left.{\left.+\left(\frac{{131}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{35}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{256}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 21.

In Corollary 20, if $\gamma \to 1$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{8}\left(\xi \left(\phi \right)+3\xi \left(\frac{2\phi +\psi }{3}\right)+3\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{1}{\psi -\phi }\underset{0}{\stackrel{1}{\int }}\xi \left(u\right)du\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{1}{2}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\frac{17}{32}\right)}^{1-{\displaystyle \frac{1}{q}}}\right.{\left(\frac{157}{1536}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{251}{1536}{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{2}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{1}{8}\right)}^{{\displaystyle \frac{1}{q}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{17}{32}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{251}{1536}{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+\frac{157}{1536}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 22.

In Corollary 17, taking $\vartheta =0$, we obtain the left-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\frac{5}{8}\right)}^{\gamma \left(1-{\displaystyle \frac{1}{q}}\right)}\right.\left(\left(\frac{{103}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{96}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{15}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{96}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q\right.\hfill \\ \hfill & {\left.+\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{5^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{96}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{3^{\gamma }}{2^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}\left(\left(\frac{{11}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{12}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q\right.\hfill \\ \hfill & {\left.+\left(\frac{7^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{12}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 23.

In Corollary 17, taking $\lambda =0$, we obtain the right-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left(\frac{3^{\gamma }}{2^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}+\frac{7^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{12}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\left(\frac{{11}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{12}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{3^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{5^{\gamma }}{8^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{5^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{96}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\left(\frac{{103}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{96}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{5\mathrm{\Gamma }\left(1+\gamma \right)}{{32}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Corollary 24.

In Corollary 17, taking $\vartheta =\lambda$, we obtain Simpson’s second formula inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{1-{\displaystyle \frac{1}{q}}}\left({\left(\frac{{17}^{\gamma }}{{32}^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \times {\left(\left(\frac{{589}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{173}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\left(\frac{{229}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{69}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1^{\gamma }}{2^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{3^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{4^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right)}^{{\displaystyle \frac{1}{q}}}{\left({\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left(\frac{{17}^{\gamma }}{{32}^{\gamma }}\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\left(\frac{{229}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{69}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\left(\frac{{589}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{{173}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}{{768}^{\gamma }\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Theorem 4.

Assume that all the assumptions of Theorem 1 are satisfied. If ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$ is generalized convex for $q>1$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left({\mathcal{M}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{M}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left({\mathcal{M}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\mathcal{M}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{N}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{N}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{N}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\mathcal{N}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{K}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{K}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+{\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left({\mathcal{K}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+{\mathcal{K}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

where ${\mathcal{M}}_i,{\mathcal{N}}_i$, and ${\mathcal{K}}_i$ are defined for $i=1,2,\dots ,5$ by (15)–(29), respectively.

Proof. 

From Lemma 6, the properties of the modulus, and improved generalized power mean inequality, along with the generalized convexity of ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times {\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\right)}^{1-{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & \times \left.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }\left({\left(1-t\right)}^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+t^{\gamma }{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right){\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill =& \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left({\mathcal{M}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{M}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left({\mathcal{M}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+{\mathcal{M}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{N}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{N}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{N}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{2\phi +\psi }{3}\right)\right|}^q+{\mathcal{N}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{K}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{K}}_2\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+{\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left({\mathcal{K}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left({\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +2\psi }{3}\right)\right|}^q+{\mathcal{K}}_5\left(\gamma ,\lambda ,\vartheta \right){\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right),\hfill \end{array}$$

where we have used

$$\begin{array}{cc}\hfill {\mathcal{M}}_1\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+2^{\gamma }{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\mathcal{M}}_2\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{2\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+2^{\gamma }{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{t}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\hfill \\ \hfill & +2^{\gamma }{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)-2^{\gamma }{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\mathcal{M}}_4\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& 2^{\gamma }{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)-{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)},\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\mathcal{M}}_5\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{2\gamma }{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& 2^{\gamma }{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right)-{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)},\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\mathcal{N}}_1\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ 2^{\gamma }{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\hfill \\ -{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill {\mathcal{N}}_2\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{2\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ 2^{\gamma }{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right)\hfill \\ +\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ \frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill {\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left(1-t\right)}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{\vartheta -5\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ \left(2^{\gamma }{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }-{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\right)\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\hfill \\ +\left(1^{\gamma }-2^{\gamma }{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\right)\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right)\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ {\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathcal{N}}_4\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ 2^{\gamma }{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\hfill \\ +\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ {\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\mathcal{N}}_5\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{2\gamma }{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \left\{\begin{array}{ccc}\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ 2^{\gamma }{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right)\hfill \\ +\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill & \mathrm{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ {\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\hfill & \mathrm{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\mathcal{K}}_1\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& 2^{\gamma }{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)-{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill {\mathcal{K}}_2\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left(1-t\right)}^{2\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& 2^{\gamma }{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right)-{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill {\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left(1-t\right)}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& \frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}-{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\hfill \\ \hfill & +2^{\gamma }{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)-2^{\gamma }{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill {\mathcal{K}}_4\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& 2^{\gamma }{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{3\gamma }\left(\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)-{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill {\mathcal{K}}_5\left(\gamma ,\lambda ,\vartheta \right)=& \frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{t}^{2\gamma }{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{\gamma }{\left(dt\right)}^{\gamma }\hfill \\ \hfill =& 2^{\gamma }{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{4\gamma }\left(\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}\right)-{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+\frac{\mathrm{\Gamma }\left(1+3\gamma \right)}{\mathrm{\Gamma }\left(1+4\gamma \right)}.\hfill \end{array}$$

(29)

The proof is complete. □

Corollary 25.

In Theorem 4, using the generalized convexity of ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left({\mathcal{M}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{3^{\gamma }{\mathcal{M}}_2\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{{\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right.\hfill \\ \hfill & +{\left({\mathcal{M}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{3^{\gamma }{\mathcal{M}}_3\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{M}}_5\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{{\mathcal{M}}_5\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{N}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{2^{\gamma }{\mathcal{N}}_2\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{{\mathcal{N}}_2\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{N}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{2^{\gamma }{\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right)+{\mathcal{N}}_5\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{{\mathcal{N}}_3\left(\gamma ,\lambda ,\vartheta \right)+2^{\gamma }{\mathcal{N}}_5\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left({\mathcal{K}}_1\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{{\mathcal{K}}_2\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{2^{\gamma }{\mathcal{K}}_2\left(\gamma ,\lambda ,\vartheta \right)+3^{\gamma }{\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +\left.{\left({\mathcal{K}}_4\left(\gamma ,\lambda ,\vartheta \right)\right)}^{1-{\displaystyle \frac{1}{q}}}{\left(\frac{{\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\phi \right)\right|}^q+\frac{2^{\gamma }{\mathcal{K}}_3\left(\gamma ,\lambda ,\vartheta \right)+3^{\gamma }{\mathcal{K}}_5\left(\gamma ,\lambda ,\vartheta \right)}{3^{\gamma }}{\left|{\xi }^{\left(\gamma \right)}\left(\psi \right)\right|}^q\right)}^{{\displaystyle \frac{1}{q}}}\right).\hfill \end{array}$$

Theorem 5.

Assume that all the assumptions of Theorem 1 are satisfied. If ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$ is generalized concave, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}\left({\left(\frac{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\mathrm{\Gamma }\left(1+p\gamma \right)}{\rm Y}\left(\gamma ,\lambda ,\vartheta \right)\right)}^{{\displaystyle \frac{1}{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +\psi }{2}\right)\right|\right.\hfill \\ \hfill & +\left.{\left(\frac{{\left(3\lambda \right)}^{p+1}+{\left(\lambda +4\vartheta \right)}^{p+1}}{{\left(4\left(\lambda +\vartheta \right)\right)}^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{5\phi +\psi }{6}\right)\right|+{\left(\frac{{\left(4\lambda +\vartheta \right)}^{p+1}+{\left(3\vartheta \right)}^{p+1}}{{\left(4\left(\lambda +\vartheta \right)\right)}^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +5\psi }{6}\right)\right|\right),\hfill \end{array}$$

where Υ is defined by (12) and $p,q>1$ with ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$.

Proof. 

From Lemma 6, properties of modulus, generalized Hölder inequality and generalized concavity of ${\left|{\xi }^{\left(\gamma \right)}\right|}^q$, we have

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }{\left(\lambda +\vartheta \right)}^{\gamma }}\left({\lambda }^{\gamma }\xi \left(\phi \right)+3^{\gamma }{\vartheta }^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }{\lambda }^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+{\vartheta }^{\gamma }\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\right.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\phi +t\frac{2\phi +\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}-t\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{2\phi +\psi }{3}+t\frac{\phi +2\psi }{3}\right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\hfill \\ \hfill & +{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|t-\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right|}^{p\gamma }{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\left.{\left(\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{0}{\stackrel{1}{\int }}{\left|{\xi }^{\left(\gamma \right)}\left(\left(1-t\right)\frac{\phi +2\psi }{3}+t\psi \right)\right|}^q{\left(dt\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{q}}}\right)\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left({\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\left({\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\right.{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{5\phi +\psi }{6}\right)\right|\hfill \\ \hfill & +{\left({\rm Y}\left(\gamma ,\lambda ,\vartheta \right)\right)}^{{\displaystyle \frac{1}{p}}}{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +\psi }{2}\right)\right|\hfill \\ \hfill & +{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\left({\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}+{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^{p+1}\right)}^{\gamma }\right)}^{{\displaystyle \frac{1}{p}}}\left.{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +5\psi }{6}\right)\right|\right)\hfill \\ \hfill =& \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}\left({\left(\frac{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}{\mathrm{\Gamma }\left(1+p\gamma \right)}{\rm Y}\left(\gamma ,\lambda ,\vartheta \right)\right)}^{{\displaystyle \frac{1}{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +\psi }{2}\right)\right|\right.\hfill \\ \hfill & +\left.{\left(\frac{{\left(3\lambda \right)}^{p+1}+{\left(\lambda +4\vartheta \right)}^{p+1}}{{\left(4\left(\lambda +\vartheta \right)\right)}^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{5\phi +\psi }{6}\right)\right|+{\left(\frac{{\left(4\lambda +\vartheta \right)}^{p+1}+{\left(3\vartheta \right)}^{p+1}}{{\left(4\left(\lambda +\vartheta \right)\right)}^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +5\psi }{6}\right)\right|\right),\hfill \end{array}$$

where we have used (11)–(13). The proof is complete. □

Corollary 26.

In Theorem 5, taking $\vartheta =0$, we obtain the left-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}\left({\left(\frac{5^{p+1}-1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +\psi }{2}\right)\right|\right.\hfill \\ \hfill & +\left.{\left(\frac{1+3^{p+1}}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{5\phi +\psi }{6}\right)\right|+\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +5\psi }{6}\right)\right|\right).\hfill \end{array}$$

Corollary 27.

In Theorem 5, taking $\lambda =0$, we obtain the right-Radau-type inequality

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}\left({\left(\frac{5^{p+1}-1}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +\psi }{2}\right)\right|\right.\hfill \\ \hfill & +\left.\left|{\xi }^{\left(\gamma \right)}\left(\frac{5\phi +\psi }{6}\right)\right|+{\left(\frac{1+3^{p+1}}{4^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +5\psi }{6}\right)\right|\right).\hfill \end{array}$$

Corollary 28.

In Theorem 5, taking $\lambda =\vartheta$, we obtain Simpson’s second formula inequality:

$$\begin{array}{cc}\hfill & \left|\frac{1}{8^{\gamma }}\left(\xi \left(\phi \right)+3^{\gamma }\xi \left(\frac{2\phi +\psi }{3}\right)+3^{\gamma }\xi \left(\frac{\phi +2\psi }{3}\right)+\xi \left(\psi \right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(\psi -\phi \right)}^{\gamma }}{}_{\phi }{I}_{\psi }^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}{\left(\frac{{\left(\psi -\phi \right)}^{\gamma }}{3^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\right)}^{{\displaystyle \frac{1}{q}}}{\left(\frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}\right)}^{{\displaystyle \frac{1}{p}}}\left({\left(\frac{1}{2}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +\psi }{2}\right)\right|\right.\hfill \\ \hfill & +\left.{\left(\frac{3^{p+1}+5^{p+1}}{8^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{5\phi +\psi }{6}\right)\right|+{\left(\frac{3^{p+1}+5^{p+1}}{8^{p+1}}\right)}^{{\displaystyle \frac{\gamma }{p}}}\left|{\xi }^{\left(\gamma \right)}\left(\frac{\phi +5\psi }{6}\right)\right|\right).\hfill \end{array}$$

4. Numerical Example and Graphical Illustration

In this section, we present a numerical example accompanied by a graphical representation to demonstrate the accuracy and validity of the theoretical results obtained.

Example 1.

Consider the function defined by

$$\xi \left(u\right)={\displaystyle \frac{\mathrm{\Gamma }(1+4\gamma )}{\mathrm{\Gamma }(1+5\gamma )}}{u}^{5\gamma },$$

which has the local fractional derivative

$${\xi }^{\left(\gamma \right)}\left(u\right)=u^{4\gamma }.$$

By selecting $\gamma =1$, $\phi =1$, and $\psi =2$, it becomes evident that the function $\left|{\xi }^{\prime }\left(u\right)\right|=u^4$ exhibits convexity over the specified interval, thereby fulfilling the fundamental assumption required by our results.

From Theorem 1, we obtain

$$\begin{array}{cc}\hfill \frac{2093\lambda +1273\vartheta }{3240(\lambda +\vartheta )}\le & \frac{1}{9}\left({\mathcal{L}}_1\left(\lambda ,\vartheta \right)+\frac{256}{81}\left({\mathcal{L}}_2\left(\lambda ,\vartheta \right)+{\mathcal{L}}_3\left(\lambda ,\vartheta \right)\right)\right.\hfill \\ \hfill & +\left.\frac{625}{81}\left({\mathcal{L}}_4\left(\lambda ,\vartheta \right)+{\mathcal{L}}_5\left(\lambda ,\vartheta \right)\right)+16{\mathcal{L}}_6\left(\gamma ,\lambda ,\vartheta \right)\right),\hfill \end{array}$$

(30)

where ${\mathcal{L}}_i$ for $i=1,2,\dots ,6$ are given by

$$\begin{array}{cc}\hfill {\mathcal{L}}_1\left(\lambda ,\vartheta \right)=& \frac{1}{3}-\frac{\lambda +4\vartheta }{8\left(\lambda +\vartheta \right)}+\frac{1}{6}{\left(\frac{\lambda +4\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{L}}_2\left(\lambda ,\vartheta \right)=& \frac{1}{3}-\frac{3\lambda }{8\left(\lambda +\vartheta \right)}+\frac{1}{3}{\left(\frac{3\lambda }{4\left(\lambda +\vartheta \right)}\right)}^3,\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{L}}_3\left(\lambda ,\vartheta \right)=& \left\{\begin{array}{ccc}\frac{5\lambda -\vartheta }{8\left(\lambda +\vartheta \right)}-\frac{1}{3}\hfill & \mathit{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ \frac{1}{3}{\left(\frac{5\lambda -\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^3+\frac{1}{3}-\frac{5\lambda -\vartheta }{8\left(\lambda +\vartheta \right)}\hfill & \mathit{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ \frac{1}{3}-\frac{5\lambda -\vartheta }{8\left(\lambda +\vartheta \right)}\hfill & \mathit{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{L}}_4\left(\lambda ,\vartheta \right)=& \left\{\begin{array}{ccc}\frac{1}{3}-\frac{5\vartheta -\lambda }{8\left(\lambda +\vartheta \right)}\hfill & \mathit{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}<0,\\ \frac{1}{3}{\left(\frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\right)}^3+\frac{1}{3}-\frac{5\vartheta -\lambda }{8\left(\lambda +\vartheta \right)}\hfill & \mathit{if}& 0\le \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}\le 1,\\ \frac{5\vartheta -\lambda }{8\left(\lambda +\vartheta \right)}-\frac{1}{3}\hfill & \mathit{if}& \frac{5\vartheta -\lambda }{4\left(\lambda +\vartheta \right)}>1,\end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{L}}_5\left(\lambda ,\vartheta \right)=& \frac{1}{3}-\frac{3\vartheta }{8\left(\lambda +\vartheta \right)}+\frac{1}{3}{\left(\frac{3\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^3\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathcal{L}}_6\left(\lambda ,\vartheta \right)=& \frac{1}{3}-\frac{4\lambda +\vartheta }{8\left(\lambda +\vartheta \right)}+\frac{1}{3}{\left(\frac{4\lambda +\vartheta }{4\left(\lambda +\vartheta \right)}\right)}^3.\hfill \end{array}$$

The left and right terms of inequality (30) are represented in two different views by Figure 1 for $\lambda ,\vartheta \in [0.1,10]$.

Figure 1. Graphical illustration of Theorem 1.

As shown in Figure 1, the numerical results support the theoretical statement of Theorem 1, confirming its validity.

5. Applications

5.1. Quadrature Formula

Let $\mathrm{\Theta }$ be the partition of the points $\phi =u_0<u_1<\dots <u_n=\psi$ of the interval $\left[\phi ,\psi \right]$, and consider the quadrature formula

$$\frac{1}{\mathrm{\Gamma }\left(\gamma +1\right)}\underset{\phi }{\stackrel{\psi }{\int }}\xi \left(u\right){\left(du\right)}^{\gamma }=\mathrm{\Omega }\left(\xi ,\mathrm{\Theta }\right)+R\left(\xi ,\mathrm{\Theta }\right),$$

where

$$\mathrm{\Omega }\left(\xi ,\mathrm{\Theta }\right)=\underset{i=0}{\sum ^{n-1}}\frac{{\left(u_{i+1}-u_i\right)}^{\gamma }}{\mathrm{\Gamma }\left(\gamma +1\right)}\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2u_i+u_{i+1}}{3}\right)+\xi \left(u_{i+1}\right)\right),$$

and $R\left(\xi ,\mathrm{\Theta }\right)$ denotes the associated approximation error.

Proposition 1.

Let $n\in \mathbb{N}$ and $\xi :\left[\phi ,\psi \right]\to {\mathbb{R}}^{\gamma }$ be a local fractional differentiable on $\left[\phi ,\psi \right]$ with $0\le \phi <\psi$ and ${\xi }^{\left(\gamma \right)}$$\in C_{\gamma }\left[\phi ,\psi \right]$. If $\left|{\xi }^{\left(\gamma \right)}\right|$ is genaralized convex, we have

$$\begin{array}{cc}\hfill \left|R\left(\xi ,\mathrm{\Theta }\right)\right|\le & \underset{i=0}{\sum ^{n-1}}\frac{{\left(u_{i+1}-u_i\right)}^{2\gamma }}{{27}^{\gamma }\mathrm{\Gamma }\left(1+\gamma \right)}\left(\left({\left(\frac{155}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+{\left(\frac{5}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(u_i\right)\right|\right.\hfill \\ \hfill & +\left.\left({\left(\frac{73}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+{\left(\frac{103}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(u_{i+1}\right)\right|\right).\hfill \end{array}$$

Proof. 

Applying Corollary 6 on the subintervals $\left[u_i,u_{i+1}\right],\left(i=0,1,\dots ,n-1\right)$ of the partition $\mathrm{\Theta }$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2u_i+u_{i+1}}{3}\right)+\xi \left(u_i\right)\right)-\frac{\mathrm{\Gamma }\left(\gamma +1\right)}{{\left(u_{i+1}-u_i\right)}^{\gamma }}{}_{u_i}{I}_{u_{i+1}}^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(u_{i+1}-u_i\right)}^{\gamma }}{{27}^{\gamma }}\left(\left({\left(\frac{155}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+{\left(\frac{5}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(u_i\right)\right|\right.\hfill \\ \hfill & +\left.\left({\left(\frac{73}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+{\left(\frac{103}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(u_{i+1}\right)\right|\right),i=1,2,\dots ,n-1.\hfill \end{array}$$

By multiplying both sides of the above inequalities by ${\displaystyle \frac{1}{\mathrm{\Gamma }\left(1+\gamma \right)}}{\left(u_{i+1}-u_i\right)}^{\gamma }$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{{\left(u_{i+1}-u_i\right)}^{\gamma }}{\mathrm{\Gamma }\left(\gamma +1\right)}\frac{1}{4^{\gamma }}\left(3^{\gamma }\xi \left(\frac{2u_i+u_{i+1}}{3}\right)+\xi \left(u_i\right)\right)-{}_{u_i}{I}_{u_{i+1}}^{\gamma }\xi \left(u\right)\right|\hfill \\ \hfill \le & \frac{{\left(u_{i+1}-u_i\right)}^{2\gamma }}{{27}^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)}\left(\left({\left(\frac{155}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+{\left(\frac{5}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(u_i\right)\right|\right.\hfill \\ \hfill & +\left.\left({\left(\frac{73}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}+{\left(\frac{103}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}\right)\left|{\xi }^{\left(\gamma \right)}\left(u_{i+1}\right)\right|\right),i=1,2,\dots ,n-1.\hfill \end{array}$$

Finally, by summing the above inequalities over all $i=0,1,\dots ,n-1$ and applying the triangle inequality, we obtain the desired result. □

5.2. Special Means

For arbitrary real numbers $\phi ,\psi$, we have the following:

The generalized arithmetic mean: $A\left(\phi ,\psi \right)={\displaystyle \frac{{\phi }^{\gamma }+{\psi }^{\gamma }}{2^{\gamma }}}$.

The generalized p-logarithmic mean: $L_p\left(\phi ,\psi \right)={\left[{\displaystyle \frac{\mathrm{\Gamma }\left(1+p\gamma \right)}{\mathrm{\Gamma }\left(1+\left(p+1\right)\gamma \right)}}\left({\displaystyle \frac{{\psi }^{\left(p+1\right)\gamma }-{\phi }^{\left(p+1\right)\gamma }}{{\left(\psi -\phi \right)}^{\gamma }}}\right)\right]}^{{\displaystyle \frac{1}{p}}}$, $\phi ,\psi \in \mathbb{R},\phi \ne \psi$, and $p\in \mathbb{Z}\setminus \left\{-1,0\right\}$.

Proposition 2.

Let $\phi ,\psi \in \mathbb{R}$ with $0<\phi <\psi$ and $n\ge 2$. Then, we have

$$\begin{array}{cc}\hfill & \left|A\left({\phi }^n,{\phi }^n,{\psi }^n\right)+A^n\left(\phi ,\phi ,\psi \right)+2^{\gamma }{A}^n\left(\phi ,\psi ,\psi \right)-4^{\gamma }\mathrm{\Gamma }\left(\gamma +1\right)L_n^n\left(\phi ,\psi \right)\right|\hfill \\ \hfill \le & \frac{4^{\gamma }{\left(\psi -\phi \right)}^{\gamma }}{9^{\gamma }}\left(\frac{\mathrm{\Gamma }\left(1+n\gamma \right)}{\mathrm{\Gamma }\left(1+\left(n-1\right)\gamma \right)}\right)\left[\left({\left(\frac{3}{4}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{1}{4}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\phi }^{\left(n-1\right)\gamma }\right.\hfill \\ \hfill & +\left({\left(\frac{29}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{5}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left(\frac{2\phi +\psi }{3}\right)}^{\left(n-1\right)\gamma }\hfill \\ \hfill & +\left({\left(\frac{31}{16}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}-{\left(\frac{7}{16}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\left(\frac{\phi +2\psi }{3}\right)}^{\left(n-1\right)\gamma }\hfill \\ \hfill & +\left.\left({\left(\frac{5}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+2\gamma \right)}{\mathrm{\Gamma }\left(1+3\gamma \right)}+{\left(\frac{3}{32}\right)}^{\gamma }\frac{\mathrm{\Gamma }\left(1+\gamma \right)}{\mathrm{\Gamma }\left(1+2\gamma \right)}\right){\psi }^{\left(n-1\right)\gamma }\right].\hfill \end{array}$$

Proof. 

This follows from Theorem 1 with $\lambda =2$ and $\vartheta =1$, applied to the function $\xi \left(u\right)=u^{n\gamma }$, where $\xi :\left(0,+\infty \right)\to {\mathbb{R}}^{\gamma }$. □

6. Conclusions

In this study, we delved into the intersection of profound mathematical concepts. Through the introduction of a novel hybrid identity that combines the strengths of Newton–Cotes and Gauss quadrature, we not only rediscovered well-known formulas but also extended our understanding to include the left- and right-Radau two-point rules, among others. Our exploration of biparametrized fractal integral inequalities for functions with generalized convex local fractional derivatives opens the door to innovative theoretical advancements and practical applications. These findings demonstrate the immense potential and versatility of our results, paving the way for further exploration in this exciting field.