New Versions of Midpoint Inequalities Based on Extended Riemann–Liouville Fractional Integrals

Abstract

This study aims to prove some midpoint-type inequalities for fractional extended Riemann–Liouville integrals. Crucial equality is proven to build new results. Using this equality, several midpoint-type inequalities are established via differentiable convex functions and the proposed extended fractional operators. To be more specific, the well-known Hölder, Jensen, and power mean integral inequalities are employed in the demonstrated inequalities. Additionally, many remarks based on specific selections of the main results are presented. Moreover, to illustrate the key conclusions, a few instances are provided.

1. Introduction

Over the past few years, many areas of mathematics have benefited greatly from the use of fractional calculus. At the same time, the conventional definitions of fractional derivatives and integrals have been somewhat expanded by more contemporary definitions. The careful investigation of the functional features of these new ideas is also a current area of study in mathematical analysis. Comprehensive analytical and numerical research on systems involving differential equations with fractional-order operators has led to the suggestion of potential scientific and technological applications [1,2,3]. The topic of fractional calculus and its applications for other scientific fields have extensively used fractional derivatives. Caputo and Riemann–Liouville derivatives were extensively used to represent complex dynamics in physics, biology, engineering, and numerous other fields with considerable success (see [4,5]). It is well-known knowledge that events in nature frequently involve memory-impacting systems. As a result, we regularly research the best nonlocal model to apply to each type of data. Additionally, other writers have looked at novel fractional general operators with separate, local, and nonlocal kernels [6,7,8].

On the other hand, a very amazing, fascinating, and captivating area of study is provided by the theory of convexity. With the help of this theory, we may create and refine the numerical tools needed to approach and research challenging mathematical topics. This theory has numerous potential applications in numerous fields of research, including information theory, coding theory, physics, engineering, optimization, and inequality theory. Additionally, convex mapping has widespread applications in pure and applied mathematics [9], mechanics [10], statistics [11], and economics [12], making it possibly the most fundamental and important mapping in the theory of mathematical inequality.

Convex analysis has had a recent surge in interest in both basic and advanced mathematics, which has been crucial to the generalizations and extensions of inequality theory. The inequality of Hermite–Hadamard is the first result of convex mappings, and has a straightforward geometric demonstration and a variety of applications, making it the most interesting inequality. For more details concerning the Hermite–Hadamard inequality, see [13,14,15]. Using Riemann–Liouville fractional integrals and convex analysis, Sarikaya et al. [16] recently proved many Hermite–Hadamard and trapezoidal inequalities. Set [17] used the Ostrowski inequality under Riemann–Liouville integrals for differentiable mappings. For differentiable and convex functions, Budak et al. [18] employed extended fractional integrals to demonstrate several variants of the Ostrowski and Simpson types. Hyder et al. [19] have recently studied a few types of fractional inequalities that are both general and trapezoidal in shape.

The purpose of this paper is to establish some midpoint-type inequalities for extended Riemann–Liouville fractional integrals. The general outline of the paper consists of five sections, including an introduction. In Section 2, the basic definitions of Riemann–Liouville fractional integral operators and extended Riemann–Liouville integral operators will be presented for building our main results. Moreover, we will give some Hermite–Hadamard and midpoint-type inequalities for these type of fractional integrals. In Section 3, we prove crucial equality in order to build new results. With the help of this equality, several midpoint-type inequalities will be established by differentiable convex functions arising from extended Riemann–Liouville integral operators. To be more precise, Hölder and power-mean inequalities, which are well-known in the literature, will be used in some of the proven inequalities. Furthermore, we will give some remarks by using special choices of the main findings. In Section 4, we will give some examples to illustrate the main results. We also show the correctness of these results by graphs. Section 5 contains some conclusions.

2. Preliminaries

Some of the relevant definitions and findings are discussed before continuing on to the primary results of the study. A function $\Theta :[\rho ,\sigma ]\to \mathbb{R}$ that meets the next inequality is deemed to be convex.

$$\Theta \left(g{h}_1+(1-g)h_2\right)\le g\Theta \left(h_1\right)+(1-g)\Theta \left(h_2\right),g\in [0,1],\mathrm{and}{h}_1,h_2\in [\rho ,\sigma ].$$

(1)

The function $\Theta$ is considered concave in shape if its additive inverse is convex. A concrete geometric representation of a convex function is provided by the next Hermite–Hadamard inequality.

Theorem 1 ([20]).

Suppose $\Theta :[\rho ,\sigma ]\to \mathbb{R}$ is a convex function; then, we gain the coming inequality

$$\Theta \left(\frac{\rho +\sigma }{2}\right)\le \frac{1}{\sigma -\rho }{\int }_{\rho }^{\sigma }\Theta \left(g\right)dg\le \frac{\Theta \left(\rho \right)+\Theta \left(\sigma \right)}{2}.$$

(2)

If Θ is concave, then (2) will passable in the counter orientation.

The next are some fractional calculus fundamentals and principles that will be employed in the existing research.

Definition 1.

[4] Let $\Theta \in L^1[\rho ,\sigma ]$, $\rho ,\sigma \in \mathbb{R}$ with $\rho <\sigma$. For $\delta >0$, the Riemann–Liouville δ-order integrals $K_{\rho +}^{\delta }\Theta$ and $K_{\sigma -}^{\delta }\Theta$ are proposed as

$$K_{\rho +}^{\delta }\Theta \left(h\right)=\frac{1}{\Gamma \left(\delta \right)}{\int }_{\rho }^h{\left(h-g\right)}^{\delta -1}\Theta \left(g\right)dg,h>\rho ,$$

(3)

$$K_{\sigma -}^{\delta }\Theta \left(h\right)=\frac{1}{\Gamma \left(\delta \right)}{\int }_h^{\sigma }{\left(g-h\right)}^{\delta -1}\Theta \left(g\right)dg,h<\sigma ,$$

(4)

respectively. Here, $\Gamma \left(\delta \right)={\int }_0^{\infty }{g}^{\delta -1}exp\left(g\right)dg$, and $K_{\rho +}^0\Theta \left(h\right)=K_{\sigma -}^0\Theta \left(h\right)=\Theta \left(h\right)$.

Jarad et al. [21] provided the upcoming fractional thorough integrals. Additionally, they supplied a few characteristics and linkages with some different fractional integrals.

Definition 2 ([21]).

Let $\delta >0$ and $\gamma \in (0,1]$. For $\Theta \in L^1[\rho ,\sigma ]$, the generalized fractional Riemann–Liouville integrals ${}^{\delta }{Z}_{\rho }^{\gamma }\Theta$ and ${}^{\delta }{Z}_{\sigma }^{\gamma }\Theta$ are defined by

$${}^{\delta }{Z}_{\rho +}^{\gamma }\Theta \left(h\right)=\frac{1}{\Gamma \left(\delta \right)}{\int }_{\rho }^h{\left(\frac{{(h-\rho )}^{\gamma }-{(g-\rho )}^{\gamma }}{\gamma }\right)}^{\delta -1}\frac{\Theta \left(g\right)}{{(g-\rho )}^{1-\gamma }}dg,h>\rho ,$$

(5)

and

$${}^{\delta }{Z}_{\sigma -}^{\gamma }\Theta \left(h\right)=\frac{1}{\Gamma \left(\delta \right)}{\int }_h^{\sigma }{\left(\frac{{(\sigma -h)}^{\gamma }-{(\sigma -g)}^{\gamma }}{\gamma }\right)}^{\delta -1}\frac{\Theta \left(g\right)}{{(\sigma -g)}^{1-\gamma }}dg,h<\sigma ,$$

(6)

respectively.

It is noteworthy that Sarikaya et al. [16] began with the following intriguing Hermite–Hadamard inequality using Riemann–Liouville integrals (3) and (4).

Theorem 2.

Assume $\Theta :[\rho ,\sigma ]\to \mathbb{R}$ is convex with $\Theta \in L_1[\rho ,\sigma ]$ and $\Theta >0$. Then, for $\delta >0$, we have

$$\begin{array}{ccc}\hfill \Theta \left({\displaystyle \frac{\rho +\sigma }{2}}\right)& \le & {\displaystyle \frac{\Gamma (\delta +1)}{2{\left(\sigma -\rho \right)}^{\delta }}}\left. [K_{{\rho }^{+}}^{\delta }\Theta \left(\sigma \right)+K_{{\sigma }^{-}}^{\delta }\Theta \left(\rho \right)\right]\hfill \\ & \le & {\displaystyle \frac{\Theta \left(\rho \right)+\Theta \left(\sigma \right)}{2}}.\hfill \end{array}$$

(7)

Additionally, Sarikaya and Yıldırım [22] provided a subsequent Hermite–Hadamard sort of inequality regarding the operators (3) and (4).

Theorem 3.

Assume $\Theta :[\rho ,\sigma ]\to \mathbb{R}$ is convex with $\Theta \in L_1[\rho ,\sigma ]$ and $\Theta >0$. Then, the subsequent fractional inequalities are valid:

$$\begin{array}{ccc}\hfill \Theta \left({\displaystyle \frac{\rho +\sigma }{2}}\right)& \le & {\displaystyle \frac{2^{\delta -1}\Gamma (\delta +1)}{{\left(\sigma -\rho \right)}^{\delta }}}\left. [K_{{\left(\frac{\rho +\sigma }{2}\right)}^{+}}^{\delta }\Theta \left(\sigma \right)+K_{{\left(\frac{\rho +\sigma }{2}\right)}^{-}}^{\delta }\Theta \left(\rho \right)\right]\hfill \\ & \le & {\displaystyle \frac{\Theta \left(\rho \right)+\Theta \left(\sigma \right)}{2}}.\hfill \end{array}$$

Set et al. [23] put forward a noteworthy Hermite–Hadamard inequality utilizing the fractional integrals in (5) and (6). This inequality is as follows:

Theorem 4.

Consider Θ to be a convex positive function from $[\rho ,\sigma ]$ to $\mathbb{R}$ with $\Theta \in L_1[\rho ,\sigma ]$ and $0\le \rho \le \sigma$. Then, the fractional integrals ${}^{\delta }{Z}_{\rho +}^{\gamma }$ and ${}^{\delta }{Z}_{\sigma -}^{\gamma }$ satisfy the following inequality.

$$\Theta \left(\frac{\rho +\sigma }{2}\right)\le \frac{\Gamma (\delta +1){\gamma }^{\delta }}{2{(\sigma -\rho )}^{\gamma \delta }}\left. [{}^{\delta }{Z}_{\rho +}^{\gamma }\Theta \left(\sigma \right)+{}^{\delta }{Z}_{\sigma -}^{\gamma }\Theta \left(\rho \right)\right]\le \frac{\Theta \left(\rho \right)+\Theta \left(\sigma \right)}{2},$$

(8)

where $Re\left(\delta \right)>0$ and $\gamma \in [0,1]$.

Moreover, Gözpınar [24] represented the Hermite–Hadamard inequality of a convex positive function involving the fractional operators (5) and (6) as follows:

Theorem 5.

Assume Θ is a convex positive function from $[\rho ,\sigma ]$ to $\mathbb{R}$ with $\Theta \in L_1[\rho ,\sigma ]$ and $0\le \rho \le \sigma$. Then, the fractional integrals ${}^{\delta }{Z}_{\rho +}^{\gamma }$ and ${}^{\delta }{Z}_{\sigma -}^{\gamma }$ fulfill the next inequality.

$$\Theta \left(\frac{\rho +\sigma }{2}\right)\le \frac{2^{\gamma \delta -1}\Gamma (\delta +1){\gamma }^{\delta }}{{(\sigma -\rho )}^{\gamma \delta }}\left. [{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)+}^{\gamma }\Theta \left(\sigma \right)+{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)-}^{\gamma }\Theta \left(\rho \right)\right]\le \frac{\Theta \left(\rho \right)+\Theta \left(\sigma \right)}{2}.$$

(9)

3. Main Outcomes

In order to construct new outcomes, we first show crucial equality. This equality will be employed to verify numerous midpoint-type inequalities through differentiable and convex functions and the extended Riemann–Liouville integrals (5) and (6).

Lemma 1.

Consider $\Theta :\left. [\rho ,\sigma \right]\to \mathbb{R}$ to be differentiable on $(\rho ,\sigma )$ and $\rho <\sigma .$ If ${\Theta }^{\prime }\in L_1\left. [\rho ,\sigma \right]$. Then, for $h\in \left. [\rho ,\sigma \right]$, the subsequent equality is valid:

$$\begin{array}{ccc}& & \frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\hfill \\ & =& \frac{{\left(h-\rho \right)}^2{\gamma }^{\delta }}{\sigma -\rho }\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)dg\hfill \\ & & -\frac{{\left(\sigma -h\right)}^2{\gamma }^{\delta }}{\sigma -\rho }\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)dg.\hfill \end{array}$$

(10)

Proof. 

Integrating the parts, the result is:

$$\begin{array}{ccc}& & M_1=\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)dg\hfill \\ & =& {\left.-{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\frac{\Theta \left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)}{h-\rho }\right|}_0^1\hfill \\ & & +\frac{\delta }{h-\rho }\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta -1}{\left(1-g\right)}^{\gamma -1}\Theta \left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)dg\hfill \\ & =& -\frac{\Theta (\rho +\sigma -h)}{\left(h-\rho \right){\gamma }^{\delta }}+\frac{\delta }{{\left(h-\rho \right)}^{\gamma \delta +1}}\underset{\rho +\sigma -h}{\overset{\sigma }{\int }}{\left(\frac{{\left(h-\rho \right)}^{\gamma }-{\left(u-(\rho +\sigma -h)\right)}^{\gamma }}{\gamma }\right)}^{\delta -1}\hfill \\ & \times & {\left(u-(\rho +\sigma -h)\right)}^{\gamma -1}\Theta \left(u\right)du\hfill \\ & =& -\frac{\Theta (\rho +\sigma -h)}{\left(h-\rho \right){\gamma }^{\delta }}+\frac{\Gamma (\delta +1)}{{\left(h-\rho \right)}^{\gamma \delta +1}}{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right).\hfill \end{array}$$

(11)

Similarly, we obtain

$$\begin{array}{ccc}\hfill M_2& =& \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)dg\hfill \\ & & \hfill \\ & =& \frac{\Theta (\rho +\sigma -h)}{\left(\sigma -h\right){\gamma }^{\delta }}-\frac{\Gamma (\delta +1)}{{\left(\sigma -h\right)}^{\gamma \delta +1}}{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right).\hfill \end{array}$$

(12)

By the equalities (11) and (12), we have

$$\begin{array}{ccc}& & \frac{{\left(h-\rho \right)}^2{\gamma }^{\delta }}{\sigma -\rho }{M}_1-\frac{{\left(\sigma -h\right)}^2{\gamma }^{\delta }}{\sigma -\rho }{M}_2\hfill \\ & =& \frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h),\hfill \end{array}$$

(13)

which wraps up the proof. □

Theorem 6.

Assume that $\Theta :\left. [\rho ,\sigma \right]\to \mathbb{R}$ is differentiable on $\left(\rho ,\sigma \right)$. If $\left|{\Theta }^{\prime }\right|$ is convex on $\left. [\rho ,\sigma \right]$, then for $h\in \left. [\rho ,\sigma \right]$, the fractional integrals ${}^{\delta }{Z}_{\rho +}^{\gamma }$ and ${}^{\delta }{Z}_{\sigma -}^{\gamma }$ satisfy the next inequality:

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\gamma \left(\sigma -\rho \right)} [\left({\left(h-\rho \right)}^2\left|{\Theta }^{\prime }\left(\sigma \right)\right|+{\left(\sigma -h\right)}^2\left|{\Theta }^{\prime }\left(\rho \right)\right|\right)B\left(\delta +1,\frac{2}{\gamma }\right)\hfill \\ & & \left.+\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)\left({\left(h-\rho \right)}^2+{\left(\sigma -h\right)}^2\right)\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|\right].\hfill \end{array}$$

(14)

Here, $B\left({\varkappa }_1,{\varkappa }_2\right)={\int }_0^1{g}^{{\varkappa }_1-1}{(1-g)}^{{\varkappa }_2-1}dg$ is the Beta integral formula.

Proof. 

By applying a modulus to the equality in Lemma 1, we obtain

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{{\left(h-\rho \right)}^2{\gamma }^{\delta }}{\sigma -\rho }\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|dg\hfill \\ & & +\frac{{\left(\sigma -h\right)}^2{\gamma }^{\delta }}{\sigma -\rho }\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)\right|dg.\hfill \end{array}$$

(15)

Considering that $\left|{\Theta }^{\prime }\right|$ is convex on $\left. [\rho ,\sigma \right]$, we have

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|dg\hfill \\ & \le & \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left. [g\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|+(1-g)\left|{\Theta }^{\prime }\left(\sigma \right)\right|\right]dg\hfill \\ & =& \frac{1}{{\gamma }^{\delta +1}}\left. [\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|+B\left(\delta +1,\frac{2}{\gamma }\right)\left|{\Theta }^{\prime }\left(\sigma \right)\right|\right].\hfill \end{array}$$

(16)

By a similar process, we can obtain

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)\right|dg\hfill \\ & \le & \frac{1}{{\gamma }^{\delta +1}}\left. [\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|+B\left(\delta +1,\frac{2}{\gamma }\right)\left|{\Theta }^{\prime }\left(\rho \right)\right|\right].\hfill \end{array}$$

(17)

The inequalities (16) and (17) are substituted in (15) to produce the objective result (14). □

Remark 1.

If we place $h=\frac{\rho +\sigma }{2}$ in Theorem 6, we achieve the inequality

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }{2}^{\gamma \delta -1}\Gamma (\delta +1)}{{\left(\sigma -\rho \right)}^{\gamma \delta }}\left. [{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)+}^{\gamma }\Theta \left(\sigma \right)+{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta \left(\frac{\rho +\sigma }{2}\right)\right|\hfill \\ & \le & \frac{\sigma -\rho }{4\gamma } [\left(\left|{\Theta }^{\prime }\left(\sigma \right)\right|+\left|{\Theta }^{\prime }\left(\rho \right)\right|\right)B\left(\delta +1,\frac{2}{\gamma }\right)\hfill \\ & & \left.+2\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)\left|{\Theta }^{\prime }\left(\frac{\rho +\sigma }{2}\right)\right|\right]\hfill \\ & \le & \frac{\sigma -\rho }{4\gamma }B\left(\delta +1,\frac{1}{\gamma }\right)\left(\left|{\Theta }^{\prime }\left(\sigma \right)\right|+\left|{\Theta }^{\prime }\left(\rho \right)\right|\right),\hfill \end{array}$$

which is offered by Gözpınar in [24] (Theorem 2).

Remark 2.

We obtain the following inequality when Theorem 6 is applied with $\gamma =1$

$$\begin{array}{ccc}& & \left|\frac{\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\delta }{Z}_{(\rho +\sigma -h)+}^{\delta }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\delta }{Z}_{(\rho +\sigma -h)-}^{\delta }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\left(\sigma -\rho \right)\left(\delta +2\right)} [\frac{{\left(h-\rho \right)}^2\left|{\Theta }^{\prime }\left(\sigma \right)\right|+{\left(\sigma -h\right)}^2\left|{\Theta }^{\prime }\left(\rho \right)\right|}{\delta +1}\hfill \\ & & \left.+\left({\left(h-\rho \right)}^2+{\left(\sigma -h\right)}^2\right)\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|\right],\hfill \end{array}$$

which is presented by Budak and Kapucu in [25] (Theorem 2.2).

Theorem 7.

Let $\Theta :\left. [\rho ,\sigma \right]\to \mathbb{R}$ be a differentiable function on $\left(\rho ,\sigma \right)$. If ${\left|{\Theta }^{\prime }\right|}^{\vartheta }$ is convex on $\left. [\rho ,\sigma \right]$ for some $\vartheta >1,$ then for $h\in \left. [\rho ,\sigma \right]$, then the coming inequality holds:

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\sigma -\rho }{\left(\frac{1}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }} [{\left(h-\rho \right)}^2{\left(\frac{{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\sigma -h\right)}^2{\left(\frac{{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}\right],\hfill \end{array}$$

(18)

where ${\varrho }^{-1}+{\vartheta }^{-1}=1.$

Proof. 

By combining Hölder’s inequality with the fact that ${\left|{\Theta }^{\prime }\right|}^{\vartheta }$ is convex, we deduce

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|dg\hfill \\ & \le & {\left(\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\varrho \delta }dg\right)}^{\frac{1}{\varrho }}{\left(\underset{0}{\overset{1}{\int }}{\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|}^{\vartheta }dg\right)}^{\frac{1}{\vartheta }}\hfill \\ & \le & {\left(\frac{1}{{\gamma }^{\varrho \delta +1}}B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }}{\left(\underset{0}{\overset{1}{\int }}\left. [g{\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+(1-g){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }\right]dg\right)}^{\frac{1}{\vartheta }}\hfill \\ & =& \frac{1}{{\gamma }^{\delta }}{\left(\frac{1}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }}{\left(\frac{{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}.\hfill \end{array}$$

(19)

Similarly, one can calculate that

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)\right|dg\hfill \\ & \le & \frac{1}{{\gamma }^{\delta }}{\left(\frac{1}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }}{\left(\frac{{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}.\hfill \end{array}$$

(20)

Hence, to obtain the wanted outcome, we plug inequalities (19) and (20) in (15). □

Remark 3.

Based on Theorem 7 with $h=\frac{\rho +\sigma }{2},$ we conclude the inequality:

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }{2}^{\gamma \delta -1}\Gamma (\delta +1)}{{\left(\sigma -\rho \right)}^{\gamma \delta }}\left. [{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)+}^{\gamma }\Theta \left(\sigma \right)+{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta \left(\frac{\rho +\sigma }{2}\right)\right|\hfill \\ & \le & \frac{\sigma -\rho }{4}{\left(\frac{1}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }}\hfill \\ & & \times \left. [{\left(\frac{3{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }}{4}\right)}^{\frac{1}{\vartheta }}+{\left(\frac{3{\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }}{4}\right)}^{\frac{1}{\vartheta }}\right]\hfill \\ & \le & \frac{\sigma -\rho }{4}{\left(\frac{4}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }}\left. [\left|{\Theta }^{\prime }\left(\rho \right)\right|+\left|{\Theta }^{\prime }\left(\sigma \right)\right|\right],\hfill \end{array}$$

(21)

which is provided by Gozpinar in [24] (Theorem 4).

Proof. 

The proof of the foremost inequality in (21) is obvious from the convexity of ${\left|{\Theta }^{\prime }\right|}^{\vartheta }$. For the second inequality, let ${\upsilon }_1=3{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta },$${\upsilon }_1^{*}={\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^q,$${\upsilon }_2={\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }$, and ${\upsilon }_2^{*}=3{\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }.$ Considering the fact that:

$$\sum _{l=1}^L{\left({\upsilon }_l+{\upsilon }_k^{*}\right)}^d\le \sum _{l=1}^L{\left({\upsilon }_l\right)}^d+\sum _{l=1}^L{\left({\upsilon }_l^{*}\right)}^d,0\le d<1,$$

the intended outcome may be easily attained. □

Remark 4.

We obtain the posterior inequality when Theorem 7 is applied with $\gamma =1$

$$\begin{array}{ccc}& & \left|\frac{\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\delta }{Z}_{(\rho +\sigma -h)+}^{\delta }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\delta }{Z}_{(\rho +\sigma -h)-}^{\delta }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\sigma -\rho }{\left(\frac{1}{\varrho \delta +1}\right)}^{\frac{1}{\varrho }} [{\left(h-\rho \right)}^2{\left(\frac{{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^q+{\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\sigma -h\right)}^2{\left(\frac{{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}\right],\hfill \end{array}$$

which is proved by Budak and Kapucu in [25] (Theorem 2.5).

Remark 5.

In Theorem 7, if we put $\gamma =\delta =1$, then Theorem 7 decreases to [26] (Theorem 4).

Theorem 8.

Let $\Theta :\left. [\rho ,\sigma \right]\to \mathbb{R}$ be a differentiable mapping on $\left(\rho ,\sigma \right)$. If ${\left|{\Theta }^{\prime }\right|}^{\vartheta }$ is convex on $\left. [\rho ,\sigma \right]$ for a fixed $\vartheta \ge 1,$ then for $h\in \left. [\rho ,\sigma \right]$, the next inequality is valid:

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\gamma \left(\sigma -\rho \right)}{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{1-\frac{1}{\vartheta }}\hfill \\ & & \times [{\left(h-\rho \right)}^2{\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^q\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\sigma -h\right)}^2{\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\rho \right)\right|}^q\right)}^{\frac{1}{\vartheta }}\right].\hfill \end{array}$$

(22)

Proof. 

Making use of the the convexity of ${\left|{\Theta }^{\prime }\right|}^{\vartheta }$ and the power mean integral inequality, we obtain

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|dg\hfill \\ & \le & {\left(\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }dg\right)}^{1-\frac{1}{\vartheta }}{\left(\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }{\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|}^{\vartheta }dg\right)}^{\frac{1}{\vartheta }}\hfill \\ & \le & {\left(\frac{1}{{\gamma }^{\delta +1}}B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{1-\frac{1}{\vartheta }}{\left(\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left. [g{\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+(1-g){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }\right]dg\right)}^{\frac{1}{\vartheta }}\hfill \\ & =& \frac{1}{{\gamma }^{\delta +1}}{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{1-\frac{1}{\vartheta }}\hfill \\ & & \times {\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}.\hfill \end{array}$$

(23)

Similarly, we also have

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\delta }\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)\right|dg\hfill \\ & \le & \frac{1}{{\gamma }^{\delta +1}}{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{1-\frac{1}{\vartheta }}\hfill \\ & & \times {\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}.\hfill \end{array}$$

(24)

By taking into account inequalities (23) and (24), in (15), we acquire the outcome (22). □

Corollary 1.

Taking $h=\frac{\rho +\sigma }{2},$ in Theorem 8, we obtain the inequality

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }{2}^{\gamma \delta -1}\Gamma (\delta +1)}{{\left(\sigma -\rho \right)}^{\gamma \delta }}\left. [{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)+}^{\gamma }\Theta \left(\sigma \right)+{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta \left(\frac{\rho +\sigma }{2}\right)\right|\hfill \\ & \le & \frac{\sigma -\rho }{4\gamma }{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{1-\frac{1}{\vartheta }}\hfill \\ & & \times [{\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\frac{\rho +\sigma }{2}\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\frac{\rho +\sigma }{2}\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\right]\hfill \\ & \le & \frac{\sigma -\rho }{4\gamma }{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{1-\frac{1}{\vartheta }}\hfill \\ & & \times [{\left(\frac{1}{2}\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }+\frac{1}{2}\left(B\left(\delta +1,\frac{1}{\gamma }\right)+B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\frac{1}{2}\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }+\frac{1}{2}\left(B\left(\delta +1,\frac{1}{\gamma }\right)+B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\right],\hfill \end{array}$$

which is presented by Gozpinar in [24] (Theorem 3).

Remark 6.

We obtain the posterior inequality when Theorem 8 is applied with $\gamma =1$

$$\begin{array}{ccc}& & \left|\frac{\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\delta }{Z}_{(\rho +\sigma -h)+}^{\delta }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\delta }{Z}_{(\rho +\sigma -h)-}^{\delta }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\left(\sigma -\rho \right)}{\left(\frac{1}{\delta +1}\right)}^{1-\frac{1}{\vartheta }}\hfill \\ & & \times [{\left(h-\rho \right)}^2{\left(\frac{1}{\delta +2}{\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+\frac{1}{\left(\delta +1\right)\left(\delta +2\right)}{\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\sigma -h\right)}^2{\left(\frac{1}{\left(\delta +1\right)\left(\delta +2\right)}{\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+\frac{1}{\delta +2}{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }\right)}^{\frac{1}{\vartheta }}\right],\hfill \end{array}$$

which is obtained by Budak and Kapucu in [25] (Theorem 2.7).

Remark 7.

If we assign $\gamma =\delta =1$ in Theorem 8, then Theorem 8 reduces to [26] (Theorem 5).

Theorem 9.

If $\Theta :\left. [\rho ,\sigma \right]\to \mathbb{R}$ is differentiable on $\left(\rho ,\sigma \right)$, $0\le \rho <\sigma$, and ${\left|{\Theta }^{\prime }\right|}^{\vartheta }$ is concave on $\left. [\rho ,\sigma \right]$ for $\vartheta >1,$ then for $h\in \left. [\rho ,\sigma \right]$, the following inequality holds:

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\sigma -\rho }{\left(\frac{1}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }}\hfill \\ & & \times \left. [{\left(h-\rho \right)}^2\left|{\Theta }^{\prime }\left(\frac{a+2\sigma -h}{2}\right)\right|+{\left(\sigma -h\right)}^2\left|{\Theta }^{\prime }\left(\frac{2\rho +\sigma -h}{2}\right)\right|\right],\hfill \end{array}$$

(25)

where ${\vartheta }^{-1}+{\varrho }^{-1}=1.$

Proof. 

Applying the Hölder inequality in (15) allows us to derive

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{{\left(h-\rho \right)}^2{\gamma }^{\delta }}{\sigma -\rho }{\left(\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\varrho \delta }dg\right)}^{\frac{1}{\varrho }}{\left(\underset{0}{\overset{1}{\int }}{\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|}^{\vartheta }dg\right)}^{\frac{1}{\vartheta }}\hfill \\ & & +\frac{{\left(\sigma -h\right)}^2{\gamma }^{\delta }}{\sigma -\rho }{\left(\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\varrho \delta }dg\right)}^{\frac{1}{\varrho }}{\left(\underset{0}{\overset{1}{\int }}{\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)\right|}^{\vartheta }dg\right)}^{\frac{1}{\vartheta }}.\hfill \end{array}$$

(26)

Since ${\left|{\Theta }^{\prime }\right|}^{\vartheta }$ is concave on $\left. [\rho ,\sigma \right]$, by using the Jensen integral inequality, we obtain

$$\begin{array}{ccc}& & \underset{0}{\overset{1}{\int }}{\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)\right|}^{\vartheta }dg\hfill \\ & \le & {\left|{\Theta }^{\prime }\left(\underset{0}{\overset{1}{\int }}\left(g\left(\rho +\sigma -h\right)+(1-g)\sigma \right)dg\right)\right|}^{\vartheta }\hfill \\ & =& {\left|{\Theta }^{\prime }\left(\frac{\rho +2\sigma -h}{2}\right)\right|}^{\vartheta },\hfill \end{array}$$

(27)

and similarly,

$$\underset{0}{\overset{1}{\int }}{\left|{\Theta }^{\prime }\left(g\left(\rho +\sigma -h\right)+(1-g)\rho \right)\right|}^{\vartheta }dg\le {\left|{\Theta }^{\prime }\left(\frac{2\rho +\sigma -h}{2}\right)\right|}^{\vartheta }.$$

(28)

On the other hand, we have

$$\underset{0}{\overset{1}{\int }}{\left(\frac{1-{\left(1-g\right)}^{\gamma }}{\gamma }\right)}^{\varrho \delta }dg=\frac{1}{{\gamma }^{\varrho \delta +1}}B\left(\varrho \delta +1,\frac{1}{\gamma }\right).$$

(29)

By substituting (27)–(29) in (26), the anticipated outcome is attained. □

Remark 8.

We obtain the posterior inequality when Theorem 9 is applied with $\gamma =1$

$$\begin{array}{ccc}& & \left|\frac{\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\delta }{Z}_{(\rho +\sigma -h)+}^{\delta }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\delta }{Z}_{(\rho +\sigma -h)-}^{\delta }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & \le & \frac{1}{\sigma -\rho }{\left(\frac{1}{\varrho \delta +1}\right)}^{\frac{1}{\varrho }}\left. [{\left(h-\rho \right)}^2\left|{\Theta }^{\prime }\left(\frac{a+2\sigma -h}{2}\right)\right|+{\left(\sigma -h\right)}^2\left|{\Theta }^{\prime }\left(\frac{2\rho +\sigma -h}{2}\right)\right|\right],\hfill \end{array}$$

which is derived by Budak and Kapucu in [25] (Theorem 2.9).

Remark 9.

Theorem 9 simplifies to [26] (Theorem 7) when $\gamma =\delta =1$ is used.

Corollary 2.

In accordance with Theorem 9’s presumptions, if $h=\frac{\rho +\sigma }{2},$ then we have the inequality

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }{2}^{\gamma \delta -1}\Gamma (\delta +1)}{{\left(\sigma -\rho \right)}^{\gamma \delta }}\left. [{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)+}^{\gamma }\Theta \left(\sigma \right)+{}^{\delta }{Z}_{\left(\frac{\rho +\sigma }{2}\right)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta \left(\frac{\rho +\sigma }{2}\right)\right|\hfill \\ & \le & \frac{\sigma -\rho }{4}{\left(\frac{1}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }}\left. [\left|{\Theta }^{\prime }\left(\frac{\rho +3\sigma }{4}\right)\right|+\left|{\Theta }^{\prime }\left(\frac{3\rho +\sigma }{4}\right)\right|\right].\hfill \end{array}$$

(30)

4. Examples

Here, we provide some examples of our main inequalities.

Example 1.

Let $\Theta :\left. [0,1\right]\to \mathbb{R}$ be a function with $\Theta \left(g\right)=g^3$. Then, ${\Theta }^{\prime }\left(g\right)=3g^2$ and $\left|{\Theta }^{\prime }\right|$ is convex on $\left. [0,1\right]$. So, we can apply Theorem 6 to this defined function. By considering $h=\frac{3}{4}$, we have

$$\begin{array}{ccc}\hfill {}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)& =& \frac{1}{\Gamma \left(\delta \right)}{\int }_{\frac{1}{4}}^1{\left(\frac{{\left(\frac{3}{4}\right)}^{\gamma }-{\left(t-\frac{1}{4}\right)}^{\gamma }}{\gamma }\right)}^{\delta -1}\frac{g^3}{{\left(g-\frac{1}{4}\right)}^{1-\gamma }}dg\hfill \\ & =& \frac{1}{{\gamma }^{\delta -1}\Gamma \left(\delta \right)}{\left(\frac{3}{4}\right)}^{\gamma \left(\delta -1\right)}{\int }_{\frac{1}{4}}^1{\left(1-{\left(\frac{4}{3}\right)}^{\gamma }{\left(g-\frac{1}{4}\right)}^{\gamma }\right)}^{\delta -1}\hfill \\ & & \times \left({\left(g-\frac{1}{4}\right)}^{\gamma +2}+\frac{3}{4}{\left(g-\frac{1}{4}\right)}^{\gamma +1}+\frac{3}{16}{\left(g-\frac{1}{4}\right)}^{\gamma }+\frac{1}{64}{\left(g-\frac{1}{4}\right)}^{\gamma -1}\right)dg\hfill \\ & =& \frac{1}{{\gamma }^{\delta }\Gamma \left(\delta \right)}{\left(\frac{3}{4}\right)}^{\gamma \delta -\gamma +1+\gamma +2}{\int }_0^1{\left(1-u\right)}^{\delta -1}{u}^{\frac{1-\gamma }{\gamma }}\left(u^{\frac{\gamma +2}{\gamma }}+u^{\frac{\gamma +1}{\gamma }}+\frac{u}{3}+\frac{u^{\frac{\gamma -1}{\gamma }}}{27}\right)du\hfill \\ & =& \frac{1}{{\gamma }^{\delta }\Gamma \left(\delta \right)}{\left(\frac{3}{4}\right)}^{\gamma \delta +3}\left. [B\left(\delta ,\frac{3}{\gamma }+1\right)+B\left(\delta ,\frac{2}{\gamma }+1\right)+\frac{1}{3}B\left(\delta ,\frac{1}{\gamma }+1\right)+\frac{1}{27\delta }\right],\hfill \end{array}$$

(31)

and similarly,

$$\begin{array}{ccc}\hfill {}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)& =& \frac{1}{\Gamma \left(\delta \right)}{\int }_0^{\frac{1}{4}}{\left(\frac{{\left(\frac{1}{4}\right)}^{\gamma }-{\left(\frac{1}{4}-g\right)}^{\gamma }}{\gamma }\right)}^{\delta -1}\frac{g^3}{{\left(\frac{1}{4}-g\right)}^{\gamma 1-\gamma }}dg\hfill \\ & =& \frac{1}{{\gamma }^{\delta -1}\Gamma \left(\delta \right)}{\left(\frac{1}{4}\right)}^{\gamma \left(\delta -1\right)}{\int }_0^{\frac{1}{4}}{\left(1-4^{\gamma }{\left(\frac{1}{4}-g\right)}^{\gamma }\right)}^{\delta -1}\hfill \\ & & \times \left(\frac{1}{64}{\left(\frac{1}{4}-g\right)}^{\gamma -1}-\frac{3}{16}{\left(\frac{1}{4}-g\right)}^{\gamma }+\frac{3}{4}{\left(g-\frac{1}{4}\right)}^{\gamma +1}-{\left(g-\frac{1}{4}\right)}^{\gamma +2}\right)dg\hfill \\ & =& \frac{1}{{\gamma }^{\delta }\Gamma \left(\delta \right)}{\left(\frac{1}{4}\right)}^{\gamma \delta -\gamma +1+\gamma +2}{\int }_0^1{\left(1-u\right)}^{\delta -1}{u}^{\frac{1-\gamma }{\gamma }}\left(u^{\frac{\gamma -1}{\gamma }}-3u+3u^{\frac{\gamma +1}{\gamma }}-u^{\frac{\gamma +2}{\gamma }}\right)du\hfill \\ & =& \frac{1}{{\gamma }^{\delta }\Gamma \left(\delta \right)}{\left(\frac{1}{4}\right)}^{\gamma \delta +3}\left. [\frac{1}{\delta }-3B\left(\delta ,\frac{1}{\gamma }+1\right)+3B\left(\delta ,\frac{2}{\gamma }+1\right)-B\left(\delta ,\frac{3}{\gamma }+1\right)\right].\hfill \end{array}$$

(32)

Then, the left-hand side of (14) becomes

$$\begin{array}{ccc}& & \left|\frac{{\gamma }^{\delta }\Gamma (\delta +1)}{\sigma -\rho }\left. [{\left(h-\rho \right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)+}^{\gamma }\Theta \left(\sigma \right)+{\left(\sigma -h\right)}^{1-\gamma \delta }{}^{\delta }{Z}_{(\rho +\sigma -h)-}^{\gamma }\Theta \left(\rho \right)\right]-\Theta (\rho +\sigma -h)\right|\hfill \\ & & \left|\delta {\left(\frac{3}{4}\right)}^4\left. [B\left(\delta ,\frac{3}{\gamma }+1\right)+B\left(\delta ,\frac{2}{\gamma }+1\right)+\frac{1}{3}B\left(\delta ,\frac{1}{\gamma }+1\right)+\frac{1}{27\delta }\right]\right.\hfill \\ & & \left.+\delta {\left(\frac{1}{4}\right)}^4\left. [\frac{1}{\delta }-3B\left(\delta ,\frac{1}{\gamma }+1\right)+3B\left(\delta ,\frac{2}{\gamma }+1\right)-B\left(\delta ,\frac{3}{\gamma }+1\right)\right]-\frac{1}{64}\right|.\hfill \end{array}$$

(33)

Additionally, the right-hand side of (14) takes the form

$$\begin{array}{ccc}& & \frac{1}{\gamma \left(\sigma -\rho \right)} [\left({\left(h-\rho \right)}^2\left|{\Theta }^{\prime }\left(\sigma \right)\right|+{\left(\sigma -h\right)}^2\left|{\Theta }^{\prime }\left(\rho \right)\right|\right)B\left(\delta +1,\frac{2}{\gamma }\right)\hfill \\ & & \left.+\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)\left({\left(h-\rho \right)}^2+{\left(\sigma -h\right)}^2\right)\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|\right]\hfill \\ & =& \frac{1}{\gamma }\left. [\frac{27}{16}B\left(\delta +1,\frac{2}{\gamma }\right)+\frac{15}{128}\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)\right]\hfill \\ & =& \frac{1}{\gamma }\left. [\frac{201}{128}B\left(\delta +1,\frac{2}{\gamma }\right)+\frac{15}{128}B\left(\delta +1,\frac{1}{\gamma }\right)\right].\hfill \end{array}$$

(34)

Then, by the inequality (14), we have

$$\begin{array}{ccc}& & \left|\delta {\left(\frac{3}{4}\right)}^4\left. [B\left(\delta ,\frac{3}{\gamma }+1\right)+B\left(\delta ,\frac{2}{\gamma }+1\right)+\frac{1}{3}B\left(\delta ,\frac{1}{\gamma }+1\right)+\frac{1}{27\delta }\right]\right.\hfill \\ & & \left.+\delta {\left(\frac{1}{4}\right)}^4\left. [\frac{1}{\delta }-3B\left(\delta ,\frac{1}{\gamma }+1\right)+3B\left(\delta ,\frac{2}{\gamma }+1\right)-B\left(\delta ,\frac{3}{\gamma }+1\right)\right]-\frac{1}{64}\right|\hfill \\ & \le & \frac{1}{\gamma }\left. [\frac{201}{128}B\left(\delta +1,\frac{2}{\gamma }\right)+\frac{15}{128}B\left(\delta +1,\frac{1}{\gamma }\right)\right].\hfill \end{array}$$

(35)

One can see the validity of (35) in Figure 1.

Example 2.

Consider the same function defined in Example 1. Let also $\varrho =\vartheta =2$ and $h=\frac{3}{4}.$ Since ${\Theta }^{\prime }\left(g\right)=3g^2,$${\left|{\Theta }^{\prime }\right|}^2$ is convex on $\left. [0,1\right]$. Therefore, we can apply Theorem 7 to this defined function. In light of these assumptions, the inequality’s (18) right-hand side becomes

$$\begin{array}{ccc}& & \frac{1}{\sigma -\rho }{\left(\frac{1}{\gamma }B\left(\varrho \delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{\varrho }} [{\left(h-\rho \right)}^2{\left(\frac{{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }\left(b\right)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\sigma -h\right)}^2{\left(\frac{{\left|{\Theta }^{\prime }\left(\rho \right)\right|}^{\vartheta }+{\left|{\Theta }^{\prime }(\rho +\sigma -h)\right|}^{\vartheta }}{2}\right)}^{\frac{1}{\vartheta }}\right]\hfill \\ & =& \left. [\frac{9}{16}{\left(\frac{2313}{512}\right)}^{\frac{1}{2}}+\frac{1}{16}{\left(\frac{9}{512}\right)}^{\frac{1}{2}}\right]{\left(\frac{1}{\gamma }B\left(2\delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{2}}.\hfill \end{array}$$

Thus, by (18) and (33), we can write

$$\begin{array}{ccc}& & \left|\delta {\left(\frac{3}{4}\right)}^4\left. [B\left(\delta ,\frac{3}{\gamma }+1\right)+B\left(\delta ,\frac{2}{\gamma }+1\right)+\frac{1}{3}B\left(\delta ,\frac{1}{\gamma }+1\right)+\frac{1}{27\delta }\right]\right.\hfill \\ & & \left.+\delta {\left(\frac{1}{4}\right)}^4\left. [\frac{1}{\delta }-3B\left(\delta ,\frac{1}{\gamma }+1\right)+3B\left(\delta ,\frac{2}{\gamma }+1\right)-B\left(\delta ,\frac{3}{\gamma }+1\right)\right]-\frac{1}{64}\right|\hfill \\ & \le & \left. [\frac{9}{16}{\left(\frac{2313}{512}\right)}^{\frac{1}{2}}+\frac{1}{16}{\left(\frac{9}{512}\right)}^{\frac{1}{2}}\right]{\left(\frac{1}{\gamma }B\left(2\delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{2}}.\hfill \end{array}$$

(36)

One can see the validity of (36) in Figure 2.

Example 3.

Under the all assumptions of Example 2 with $\vartheta =2,$ we have the right-hand side of (22) as:

$$\begin{array}{ccc}& & \frac{1}{\gamma \left(\sigma -\rho \right)}{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{1-\frac{1}{\vartheta }}\hfill \\ & & \times [{\left(h-\rho \right)}^2{\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\sigma \right)\right|}^q\right)}^{\frac{1}{\vartheta }}\hfill \\ & & \left.+{\left(\sigma -h\right)}^2{\left(\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right){\left|{\Theta }^{\prime }\left(\rho +\sigma -h\right)\right|}^{\vartheta }+B\left(\delta +1,\frac{2}{\gamma }\right){\left|{\Theta }^{\prime }\left(\rho \right)\right|}^q\right)}^{\frac{1}{\vartheta }}\right]\hfill \\ & =& \frac{1}{\gamma }{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{2}} [\frac{9}{16}{\left(\frac{9}{256}B\left(\delta +1,\frac{1}{\gamma }\right)+\frac{2295}{256}B\left(\delta +1,\frac{2}{\gamma }\right)\right)}^{\frac{1}{2}}\hfill \\ & & \left.+\frac{1}{16}{\left(\frac{9}{256}\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)\right)}^{\frac{1}{2}}\right]\hfill \\ & =& \frac{3}{256\gamma }{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{2}} [9{\left(B\left(\delta +1,\frac{1}{\gamma }\right)+255B\left(\delta +1,\frac{2}{\gamma }\right)\right)}^{\frac{1}{2}}\hfill \\ & & \left.+{\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)}^{\frac{1}{2}}\right].\hfill \end{array}$$

(37)

By the inequality (22), we can write

$$\begin{array}{ccc}& & \left|\delta {\left(\frac{3}{4}\right)}^4\left. [B\left(\delta ,\frac{3}{\gamma }+1\right)+B\left(\delta ,\frac{2}{\gamma }+1\right)+\frac{1}{3}B\left(\delta ,\frac{1}{\gamma }+1\right)+\frac{1}{27\delta }\right]\right.\hfill \\ & & \left.+\delta {\left(\frac{1}{4}\right)}^4\left. [\frac{1}{\delta }-3B\left(\delta ,\frac{1}{\gamma }+1\right)+3B\left(\delta ,\frac{2}{\gamma }+1\right)-B\left(\delta ,\frac{3}{\gamma }+1\right)\right]-\frac{1}{64}\right|\hfill \\ & \le & \frac{3}{256\gamma }{\left(B\left(\delta +1,\frac{1}{\gamma }\right)\right)}^{\frac{1}{2}} [9{\left(B\left(\delta +1,\frac{1}{\gamma }\right)+255B\left(\delta +1,\frac{2}{\gamma }\right)\right)}^{\frac{1}{2}}\hfill \\ & & \left.+{\left(B\left(\delta +1,\frac{1}{\gamma }\right)-B\left(\delta +1,\frac{2}{\gamma }\right)\right)}^{\frac{1}{2}}\right].\hfill \end{array}$$

(38)

One can see the validity of (38) in Figure 3.

5. Conclusions

This work discussed some midpoint-type inequalities in the frame of the extended Riemann–Liouville fractional integrals. Foremost, an equality containing the operators has been proven to make the new results. Employing this equality and the features of the differentiable convex functions, numerous midpoint-type inequalities have been proven for the extended Riemann–Liouville fractional integrals. It is remarkable that the recognized H older, Jensen, and power mean integral inequalities have been utilized in the proofs of the obtained inequalities. According to the given remarks, one can conclude that all the acquired results can be abbreviated to some inequalities existing in the literature by picking particular values of the factors. Additionally, the proof techniques followed in this work can be employed to investigate different fractional integral inequalities. Furthermore, some examples have been provided to exhibit and support the outcomes.