New Quantum Mercer Estimates of Simpson–Newton-like Inequalities via Convexity

Abstract

Recently, developments and extensions of quadrature inequalities in quantum calculus have been extensively studied. As a result, several quantum extensions of Simpson’s and Newton’s estimates are examined in order to explore different directions in quantum studies. The main motivation of this article is the development of variants of Simpson–Newton-like inequalities by employing Mercer’s convexity in the context of quantum calculus. The results also give new quantum bounds for Simpson–Newton-like inequalities through Hölder’s inequality and the power mean inequality by employing the Mercer scheme. The validity of our main results is justified by providing examples with graphical representations thereof. The obtained results recapture the discoveries of numerous authors in quantum and classical calculus. Hence, the results of these inequalities lead us to the development of new perspectives and extensions of prior results.

1. Introduction

Integral inequalities have historically been viewed as a classical field of research. From classical to contemporary applications, inequalities have been used in mathematical analysis. In 1934, Polya and Hardy introduced classical work on inequalities. Integral inequalities play a vital role in differential equation theory. Many researchers have studied integral inequalities in classical calculus, along with their applications (see [1,2]). Because the value of mathematical inequalities was well established in the past, inequalities such as the Hermite–Hadamard, Hölder, Ostrowski, Jensen, Hardy, and Cauchy–Schwartz inequalities have played an essential role in the theory of classical calculus and quantum calculus [3].

Convexity is a growing area of research that has applications in complex analysis, number theory, and many other fields. Convexity also has a significant impact on people’s lives with its numerous applications [1,3]. Convex functions are defined as follows.

Definition 1

([1]). Let $\mathfrak{g}:[{\rho }_1,{\rho }_2]\subseteq \Re \to \Re$ be convex if, for every $\mathsf{\varkappa },\mathrm{y}\in [{\rho }_1,{\rho }_2]$ and every $\lambda \in [0,1]$, we have:

$$\begin{array}{c}\hfill \mathfrak{g}(\lambda \mathrm{y}+(1-\lambda \left)\mathsf{\varkappa }\right)\le \lambda \mathfrak{g}\left(\mathrm{y}\right)+(1-\lambda )\mathfrak{g}\left(\mathsf{\varkappa }\right).\end{array}$$

which holds for every $\mathsf{\varkappa },\mathrm{y}\in [{\rho }_1,{\rho }_2]$ and every $\lambda \in [0,1].$

Convexity theory also provides us with an amazing framework for initiating and developing numerical tools to tackle and study complicated problems in mathematics. Due to the number of expedient properties, they are magical, especially in optimization theory. The theories of mathematical inequalities and convex functions have a wonderful relationship. One of the most remarkable inequalities, which, we may say, is the natural extension of the convex function $\mathfrak{g}:[{\wp }_1,{\wp }_2]\to \Re$, is Jensen’s inequality [1], which is given as:

$$\mathfrak{g}\left(\underset{j=1}{\sum ^{\mathcal{N}}}{\omega }_j{\mathsf{\varkappa }}_j\right)\le \left(\underset{j=1}{\sum ^{\mathcal{N}}}{\omega }_j\mathfrak{g}\left({\mathsf{\varkappa }}_j\right)\right),$$

(1)

for all ${\mathsf{\varkappa }}_j\in [{\wp }_1,{\wp }_2],$${\omega }_j\in \left[0,1\right]$ satisfying $\sum {\omega }_j=1$ for $\left(j=1,2,\dots ,\mathcal{N}\right)$. For $\mathcal{N}=2$, Jensen’s inequality (1) recaptures the definition of a convex function. There are several important applications of Jensen’s inequality in economics, finance, statistics, and optimization, but it is also the most effective inequality for predicting the estimations of the bounds of distance functions in information theory (see [4,5]).

In the year 2003, Mercer [6] introduced an interesting variant of Jensen’s inequality, which is called the Jensen–Mercer inequality:

$$\mathfrak{g}\left({\wp }_1+{\wp }_2-\underset{j=1}{\sum ^{\mathcal{N}}}{\omega }_j{\mathsf{\varkappa }}_j\right)\le \mathfrak{g}\left({\wp }_1\right)+\mathfrak{g}\left({\wp }_2\right)-\underset{j=1}{\sum ^{\mathcal{N}}}{\omega }_j\mathfrak{g}\left({\mathsf{\varkappa }}_j\right),$$

(2)

which holds for all finite positive increasing sequences ${\mathsf{\varkappa }}_j\in \left[{\wp }_1,{\wp }_2\right],$ for $\left(j=1,2,\dots ,j\right)$, along with the weights ${\omega }_j\in \left[0,1\right]$ defined in (1). Many scholars have investigated and studied the Jensen–Mercer inequality over the years, and they have even brought it to higher dimensions, using it for convex operators along with several purifications, operator variants for super-quadratic functions, improvements, and many generalizations with applications in information theory (see [7,8,9,10]).

In the applied sciences, there are a number of significant inequalities due to their important perspectives. However, the present study revolves around the renowned Simpson inequality [11], whose error estimates are written as:

$$\begin{array}{c}\hfill \left|\frac{1}{3}\left\{\frac{\mathfrak{g}\left({\wp }_1\right)+\mathfrak{g}\left({\wp }_2\right)}{2}+2\mathfrak{g}\left(\frac{{\wp }_1+{\wp }_2}{2}\right)-\frac{1}{{\wp }_2-{\wp }_1}{\int }_{{\wp }_1}^{{\wp }_2}\mathfrak{g}\left(\mathsf{\varkappa }\right)d\mathsf{\varkappa }\right\}\right|\\ \hfill \le \frac{1}{2880}{\left|\left|{\mathfrak{g}}^4\right|\right|}_{\infty }{\left({\wp }_2-{\wp }_1\right)}^4,\end{array}$$

where $\mathfrak{g}:[{\wp }_1,{\wp }_2]\to \Re$ is a four-time continuously differentiable mapping on $({\wp }_1,{\wp }_2)$ and ${\left|\left|{\mathfrak{g}}^4\right|\right|}_{\infty }={sup}_{\mathsf{\varkappa }\in \left({\wp }_1,{\wp }_2\right)}\left|{\mathfrak{g}}^4\left(\mathsf{\varkappa }\right)\right|<\infty .$

Simpson-type inequalities are a topic of great interest for many scholars, and they have been explored and studied for various classes of functions. Some fractional Simpson results were established with applications in [12,13].

One of the trickiest math problems to comprehend is in the history of quantum calculus, which is sometimes known as $\widehat{\mathrm{q}}$-calculus or $\widehat{\mathrm{q}}$-disease and dates back 300 years to Bernoulli and Euler. When there is no limit in calculus, it is referred as $\widehat{\mathrm{q}}$-calculus. Euler was the creator of $\widehat{\mathrm{q}}$-calculus and the inventor of the $\widehat{\mathrm{q}}$-parameter by using the $\widehat{\mathrm{q}}$-parameter in Newton’s work on infinite series. Jackson was the first to present the developing ideas about $\widehat{\mathrm{q}}$-calculus in a symmetrical manner in the nineteenth century by introducing $\widehat{\mathrm{q}}$-definite integrals [14]. Al-Salam presented a fractional integral operator and a $\widehat{\mathrm{q}}$-analog of the Riemann–Liouville fractional integral operator from 1966 to 1967 [15]. Due to its applicability in a wide range of domains, including mathematics, mechanics, and physics, there is now a tremendous rise in the area of $\widehat{\mathrm{q}}$-calculus. One can see this in [15,16,17] and the references therein. $\widehat{\mathrm{q}}$-difference operators are of huge importance because of their applications in a variety of mathematical disciplines, including orthogonal polynomials, basic hypergeometric functions, relativity, combinatorics, and mechanics. Many essential concepts of quantum calculus are covered in Kac and Cheung’s [18] book. These ideas help us to develop new inequalities, which can be useful in the discovery of new boundaries. The following are some of the fundamental definitions of quantum calculus.

2. Preliminaries

We first present the definitions of quantum derivatives and quantum integrals.

$$\begin{array}{c}\hfill {\left[n\right]}_{\widehat{\mathrm{q}}}=\frac{{\widehat{\mathrm{q}}}^n-1}{\widehat{\mathrm{q}}-1}.\end{array}$$

The ${}_{{\rho }_1}{D}_{\widehat{\mathrm{q}}}$-difference operator and ${\widehat{\mathrm{q}}}_{{\rho }_1}$-integral were first presented by Tariboon in [19]. Later, Bermudo et al. [20] intervened with the concept of the ${\widehat{\mathrm{q}}}^{{\rho }_2}$-derivative and ${\widehat{\mathrm{q}}}^{{\rho }_2}$-integral.

Definition 2

([20]). The ${\widehat{\mathrm{q}}}^{{\rho }_2}$-derivative of mapping $\mathfrak{g}:[{\rho }_1,{\rho }_2]\to \mathbb{R}$ is defined as:

$${}^{{\rho }_2}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)=\frac{\mathfrak{g}(\widehat{\mathrm{q}}\mathsf{\varkappa }+(1-\widehat{\mathrm{q}}){\rho }_2)-\mathfrak{g}\left(\mathsf{\varkappa }\right)}{(1-\widehat{\mathrm{q}})({\rho }_2-\mathsf{\varkappa })},\mathrm{if}\mathsf{\varkappa }\ne {\rho }_2.$$

$${\displaystyle \mathrm{If}\mathsf{\varkappa }={\rho }_2,{}^{{\rho }_2}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)=\underset{\mathsf{\varkappa }\to {\rho }_2}{lim}{}^{{\rho }_2}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right),}$$

which is a $\widehat{\mathrm{q}}$-Jackson derivative.

Definition 3

([20]). The ${\widehat{\mathrm{q}}}^{{\rho }_2}$-integral of mapping $\mathfrak{g}:[{\rho }_1,{\rho }_2]\to \mathbb{R}$ is defined as:

$${\int }_{{\rho }_1}^{{\rho }_2}\mathfrak{g}\left(\mathsf{\varkappa }\right){}^{{\rho }_2}{d}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }=(1-\widehat{\mathrm{q}})({\rho }_2-{\rho }_1)\sum _{n=0}^{\infty }{\widehat{\mathrm{q}}}^n\mathfrak{g}({\widehat{\mathrm{q}}}^n{\rho }_1+(1-{\widehat{\mathrm{q}}}^n){\rho }_2),$$

which is a $\widehat{\mathrm{q}}$-Jackson integral.

Theorem 1

([20]). If $\mathfrak{g}:[{\rho }_1,{\rho }_2]\to \mathbb{R}$ is a convex mapping that is differentiable on $[{\rho }_1,{\rho }_2]$, then the following inequality is true:

$$\mathfrak{g}\left(\frac{{\rho }_1+\widehat{\mathrm{q}}{\rho }_2}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)\le \frac{1}{{\rho }_2-{\rho }_1}{\int }_{{\rho }_1}^{{\rho }_2}\mathfrak{g}\left(\mathsf{\varkappa }\right){}^{{\rho }_2}{d}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }\le \frac{\mathfrak{g}\left({\rho }_1\right)+\widehat{\mathrm{q}}\mathfrak{g}\left({\rho }_2\right)}{{\left[2\right]}_{\widehat{\mathrm{q}}}},$$

where $\widehat{\mathrm{q}}\in (0,1)$.

Lemma 1.

We have the equality

$${\int }_{{\rho }_1}^{{\rho }_2}{(\mathsf{\varkappa }-{\rho }_1)}^{\alpha }{}_{{\rho }_1}{d}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }=\frac{{({\rho }_2-{\rho }_1)}^{\alpha +1}}{{[\alpha +1]}_{\widehat{\mathrm{q}}}},$$

for $\alpha \in \mathbb{R}\backslash \{-1\}$.

It is pertinent to mention an important lemma of interest.

Lemma 2

([21]). For the continuous functions $\mathfrak{g},\mathfrak{h}:[{\rho }_1,{\rho }_2]\to \mathbb{R}$, the following inequality is true:

$$\begin{array}{cc}\hfill {\int }_0^c\mathfrak{h}\left(\eta \right){}^{{\rho }_2}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}(\eta {\rho }_1+(1-\eta ){\rho }_2)d_{\widehat{\mathrm{q}}}\eta & =\frac{1}{{\rho }_2-{\rho }_1}{\int }_0^c{D}_{\widehat{\mathrm{q}}}\mathfrak{h}\left(\eta \right)\mathfrak{g}(\widehat{\mathrm{q}}\eta {\rho }_1+(1-\widehat{\mathrm{q}}\eta ){\rho }_2)d_{\widehat{\mathrm{q}}}\eta \hfill \\ \hfill & -\frac{\mathfrak{h}\left(\eta \right)\mathfrak{g}(\eta {\rho }_1+(1-\eta ){\rho }_2)}{{\rho }_2-{\rho }_1}{|}_0^c.\hfill \end{array}$$

Despite our concentration on the thrill and romance of the development of quantum calculus and its applications in specific areas of the mathematical sciences and physics, $\widehat{\mathrm{q}}$-analogs of integral inequalities are a topic of supreme interest. Using the ${}_{{\rho }_1}{D}_{\widehat{\mathrm{q}}}$-derivative, ${\widehat{\mathrm{q}}}_{{\rho }_1}$-integral, ${\widehat{\mathrm{q}}}^{{\rho }_2}$-derivative, and ${\widehat{\mathrm{q}}}^{{\rho }_2}$-integral, several integral inequalities—namely, those of Hölder, Grüss, Ostrwoski, Hermite–Hadamrd, the trapezoid, and the midpoint —have been investigated with respect to various kinds of functions (see [20,22,23,24,25,26]). Some quantum integral inequalities in coordinates can be observed in [27,28]. New quantum versions of Simpson- and Newton-type inequalities relevant to convex functions were developed in [29,30,31], and their significance was established. Recently, Budak et al. introduced a variant of the quantum Hermite–Jensen–Mercer inequalities [32,33], while in [34], Bohner et al. provided their post-quantum extensions.

Motivated by the ongoing trend, the purpose of this study is to give an analysis of quantum Simpon-like inequalities under Mercer’s concept. We formulated new quantum auxiliary results. By applying convexity and the related Jensen–Mercer inequality, we give a variety of new estimates of Simpson’s quantum inequality.

3. Auxiliary Results

In this section, we describe our discovery of novel quantum Simpson–Mercer and quantum Newton–Mercer-like identities using ${\widehat{\mathrm{q}}}^{{\rho }_2}$-integrals.

Lemma 3.

Let a mapping $\mathfrak{g}:\mathcal{I}=[{\rho }_1,{\rho }_2]\subseteq {\Re }^{+}\to \Re$ be differentiable on $({\rho }_1,{\rho }_2)$. If ${}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\in \mathcal{L}[{\rho }_1,{\rho }_2]$, then, for all $\mathsf{\varkappa },\mathrm{y}\in [{\rho }_1,{\rho }_2]$ with $\mathsf{\varkappa }\le \mathrm{y}$, the following identity holds:

$$\begin{array}{cc}\hfill & \frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]\hfill \\ \hfill & =\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\int }_0^1\wp \left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))d_{\widehat{\mathrm{q}}}\lambda ,\hfill \end{array}$$

where

$$\wp \left(\lambda \right)=\left\{\begin{array}{cc}\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}},\hfill & \lambda \in \left[0,\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}},\hfill & \lambda \in \left[\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}},1\right].\hfill \end{array}\right.$$

Proof. 

Taking into account the fundamental properties of quantum integrals, we have

$$\begin{array}{cc}\hfill & {\int }_0^1\wp \left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & ={\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\left(\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right)d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\left(\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right)d_{\widehat{\mathrm{q}}}\lambda ,\hfill \\ \hfill & =\frac{\widehat{\mathrm{q}}{\left[4\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +{\int }_0^1\left(\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & =I_1+I_2.\hfill \end{array}$$

Now, as a consequence of Lemma 2, we get

$$\begin{array}{cc}\hfill & I_1=\frac{\widehat{\mathrm{q}}{\left[4\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & =-\frac{\widehat{\mathrm{q}}{\left[4\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\frac{\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))}{\mathrm{y}-\mathsf{\varkappa }}{|}_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =-\frac{\widehat{\mathrm{q}}{\left[4\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\left[\mathfrak{g}\left({\rho }_1+{\rho }_2-\left(\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathrm{y}+\left(1-\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)\mathsf{\varkappa }\right)\right)-\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]\hfill \\ \hfill & =-\frac{\widehat{\mathrm{q}}{\left[4\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\left[\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)-\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right].\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill I_2& ={\int }_0^1{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\left(\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right)d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & =-\left(\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right)\frac{\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))}{\mathrm{y}-\mathsf{\varkappa }}{|}_0^1\hfill \\ \hfill & +\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_0^1{D}_{\widehat{\mathrm{q}}}(\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}})\mathfrak{g}({\rho }_1+{\rho }_2-(\widehat{\mathrm{q}}\lambda \mathrm{y}+(1-\widehat{\mathrm{q}}\lambda )\mathsf{\varkappa }))d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & =-\left(1-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right)\frac{\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})}{\mathrm{y}-\mathsf{\varkappa }}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\frac{\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })}{\mathrm{y}-\mathsf{\varkappa }}\hfill \\ \hfill & +\frac{1}{\mathsf{\varkappa }-\mathrm{y}}{\int }_0^1\mathfrak{g}({\rho }_1+{\rho }_2-(\widehat{\mathrm{q}}\lambda \mathrm{y}+(1-\widehat{\mathrm{q}}\lambda )\mathsf{\varkappa }))d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & =-\frac{{\left[6\right]}_{\widehat{\mathrm{q}}}-{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\frac{\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})}{\mathrm{y}-\mathsf{\varkappa }}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\frac{\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })}{\mathrm{y}-\mathsf{\varkappa }}\hfill \\ \hfill & +\frac{(1-\widehat{\mathrm{q}}){\mathsf{\Sigma }}_{n=0}^{\infty }{\widehat{\mathrm{q}}}^n\mathfrak{g}({\rho }_1+{\rho }_2-({\widehat{\mathrm{q}}}^{n+1}\mathrm{y}+(1-{\widehat{\mathrm{q}}}^{n+1})\mathsf{\varkappa }))}{\mathrm{y}-\mathsf{\varkappa }}\hfill \\ \hfill & =-\frac{{\widehat{\mathrm{q}}}^5}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\hfill \\ \hfill & +\frac{(1-\widehat{\mathrm{q}}){\mathsf{\Sigma }}_{n=0}^{\infty }{\widehat{\mathrm{q}}}^{n+1}\mathfrak{g}({\rho }_1+{\rho }_2-({\widehat{\mathrm{q}}}^{n+1}\mathrm{y}+(1-{\widehat{\mathrm{q}}}^{n+1})\mathsf{\varkappa }))}{\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })}\hfill \\ \hfill & =-\frac{{\widehat{\mathrm{q}}}^5}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\hfill \\ \hfill & -\frac{(1-\widehat{\mathrm{q}})}{\widehat{\mathrm{q}}(\mathsf{\varkappa }-\mathrm{y})}\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{1}{\widehat{\mathrm{q}}{(\mathsf{\varkappa }-\mathrm{y})}^2}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & =-\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}(\mathrm{y}-\mathsf{\varkappa })}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\hfill \\ \hfill & +\frac{1}{\widehat{\mathrm{q}}{(\mathsf{\varkappa }-\mathrm{y})}^2}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda .\hfill \end{array}$$

Then, it follows that

$$\begin{array}{cc}\hfill \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })\left(I_1+I_2\right)& =\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right].\hfill \end{array}$$

Thus, the required equality is captured here. □

Remark 1.

If we set $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ in Lemma 3, it can be reduced to the following equality:

$$\begin{array}{ccc}& & \frac{1}{{\rho }_2-{\rho }_1}{\int }_{{\rho }_1}^{{\rho }_2}\mathfrak{g}\left(\lambda \right){}^{{\rho }_2}{d}_{\widehat{\mathrm{q}}}\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}\left({\rho }_1\right)+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\frac{{\rho }_1+\widehat{\mathrm{q}}{\rho }_2}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}\left({\rho }_2\right)\right]\hfill \\ & =& \widehat{\mathrm{q}}({\rho }_2-{\rho }_1){\int }_0^1\wp \left(\lambda \right){}^{{\rho }_2}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}(\lambda {\rho }_1+(1-\lambda ){\rho }_2)d_{\widehat{\mathrm{q}}}\lambda ,\hfill \end{array}$$

which was proven in [31].

Lemma 4.

With the suppositions of Lemma 3, we have the following identity:

$$\begin{array}{cc}\hfill & \frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{1+\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+(1+\widehat{\mathrm{q}})\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{(1+\widehat{\mathrm{q}})}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{(1+\widehat{\mathrm{q}})\mathrm{y}+{\widehat{\mathrm{q}}}^2\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })]\hfill \\ \hfill & =\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\int }_0^1\psi \left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })d_{\widehat{\mathrm{q}}}\lambda ,\hfill \end{array}$$

(3)

where

$$\psi \left(\lambda \right)=\left\{\begin{array}{cc}\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}},\hfill & \lambda \in \left[0,\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \lambda -\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}},\hfill & \lambda \in \left[\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}},\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \lambda -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}},\hfill & \lambda \in \left[\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}},1\right].\hfill \end{array}\right.$$

Proof. 

By using the fundamental properties of quantum integrals, we have

$$\begin{array}{cc}\hfill & {\int }_0^1\psi \left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill =& {\int }_0^{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })\left(\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}\right)d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill +& {\int }_{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })\left(\lambda -\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill +& {\int }_{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^1{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })\left(\lambda -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}\right)d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill =& I_1+I_2+I_3.\hfill \end{array}$$

The desired results may be attained if steps similar to those in Lemma 3 are used for the rest of this proof. □

Remark 2.

If we set $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ in Lemma 4, it can be reduced to the following equality:

$$\begin{array}{cc}\hfill & \frac{1}{{\rho }_2-{\rho }_1}{\int }_{{\rho }_1}^{{\rho }_2}\mathfrak{g}\left(\lambda \right){}^{{\rho }_2}{d}_{\widehat{\mathrm{q}}}\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}\left({\rho }_1\right)+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left(\frac{{\rho }_1+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}{\rho }_2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{(1+\widehat{\mathrm{q}})}\mathfrak{g}\left(\frac{(1+\widehat{\mathrm{q}}){\rho }_1+{\widehat{\mathrm{q}}}^2{\rho }_2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}\left({\rho }_2\right)]\hfill \\ \hfill & =\widehat{\mathrm{q}}({\rho }_2-{\rho }_1){\int }_0^1\psi \left(\lambda \right){}^{{\rho }_2}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}(\lambda {\rho }_1+(1-\lambda ){\rho }_2)d_{\widehat{\mathrm{q}}}\lambda ,\hfill \end{array}$$

which was proven in [31].

3.1. Simpson–Mercer $1/3$ Formula-Type Inequalities

Theorem 2.

Let the assumptions of Lemma 3 hold. If $\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\right|$ is convex on $[{\rho }_1,{\rho }_2]$, then the following inequality holds:

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })\left\{\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)|+|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\left|\right(\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+2\widehat{\mathrm{q}}\frac{(1+{\left[5\right]}_{\widehat{\mathrm{q}}}^2)}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\hfill \\ \hfill & -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3})-|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)|\left[{\mathfrak{c}}_{1\widehat{\mathrm{q}}}+{\mathfrak{c}}_{2\widehat{\mathrm{q}}}\right]-|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\left|[{\mathrm{d}}_{1\widehat{\mathrm{q}}}+{\mathrm{d}}_{2\widehat{\mathrm{q}}}]\right\},\hfill \end{array}$$

(4)

where

$$\begin{array}{cc}\hfill {\omega }_{1\widehat{\mathrm{q}}}& ={\left[3\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^3{\left[2\right]}_{\widehat{\mathrm{q}}}^3,\hfill \\ \hfill {\omega }_{2\widehat{\mathrm{q}}}& ={\left[3\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^3{\left[2\right]}_{\widehat{\mathrm{q}}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathfrak{c}}_{1\widehat{\mathrm{q}}}=& {\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\lambda |\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =\frac{2{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}^2+{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}}{{\omega }_{1\widehat{\mathrm{q}}}},\hfill \\ \hfill {\mathfrak{c}}_{2\widehat{\mathrm{q}}}=& {\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}(1-\lambda )|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =\frac{2{\widehat{\mathrm{q}}}^2{\left[5\right]}_{\widehat{\mathrm{q}}}^3{\left[2\right]}_{\widehat{\mathrm{q}}}^2+{\left[6\right]}_{\widehat{\mathrm{q}}}^3(1+{\left[2\right]}_{\widehat{\mathrm{q}}}^3)-{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2(1+{\left[2\right]}_{\widehat{\mathrm{q}}}^2)}{{\omega }_{1\widehat{\mathrm{q}}}},\hfill \\ \hfill {\mathrm{d}}_{1\widehat{\mathrm{q}}}=& {\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1\lambda |\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =\left(\frac{\widehat{\mathrm{q}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2}-\frac{\widehat{\mathrm{q}}\left(\widehat{\mathrm{q}}+2\right)}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+2\frac{\widehat{\mathrm{q}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}-{\widehat{\mathrm{q}}}^2}{{\omega }_{2\widehat{\mathrm{q}}}}\right),\hfill \\ \hfill {\mathrm{d}}_{2\widehat{\mathrm{q}}}=& {\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1(1-\lambda )|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =2\frac{\widehat{\mathrm{q}}{\left[5\right]}_{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}-{\widehat{\mathrm{q}}}^2{\left[5\right]}_{\widehat{\mathrm{q}}}^3}{{\omega }_{2\widehat{\mathrm{q}}}}+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}}\hfill \\ \hfill & -\left[\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}(2\widehat{\mathrm{q}}+{\widehat{\mathrm{q}}}^2)-\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}}\right],\hfill \end{array}$$

and $0<\widehat{\mathrm{q}}<1$.

Proof. 

By taking the modulus in Lemma 3, we have

$$\begin{array}{cc}\hfill & |\frac{1}{(\mathrm{y}-\mathsf{\varkappa })}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\right|d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\right|d_{\widehat{\mathrm{q}}}\lambda .\hfill \end{array}$$

By using the convexity of ${|}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}|$, we get

$$\begin{array}{cc}\hfill & \le {\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|\left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|\right]-\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\lambda |\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill -& \left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}(1-\lambda )|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_q\mathfrak{g}\left({\rho }_2\right)\right|\right]-\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|\lambda d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1(1-\lambda )|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda .\hfill \end{array}$$

Here, we have

$${\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\left|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right|d_{\widehat{\mathrm{q}}}\lambda =\frac{2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}+\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3},$$

and by using similar operations, we have

$$\begin{array}{cc}\hfill & {\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1\left|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\right|d_{\widehat{\mathrm{q}}}\lambda =-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+2\widehat{\mathrm{q}}\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}^2}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\hfill \\ & =\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })\left\{\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)|+|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\left|\right(\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\hfill \\ \hfill & -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+2\widehat{\mathrm{q}}\frac{(1+{\left[5\right]}_{\widehat{\mathrm{q}}}^2)}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2})-|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)|\left[{\mathfrak{c}}_{1\widehat{\mathrm{q}}}+{\mathfrak{c}}_{2\widehat{\mathrm{q}}}\right]-|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\left|[{\mathrm{d}}_{1\widehat{\mathrm{q}}}+{\mathrm{d}}_{2\widehat{\mathrm{q}}}]\right\}.\hfill \end{array}$$

Hence, the proof is completed. □

We give the following example to show the validity of the results given in Theorem 2.

Example 1.

Let us consider the function $\mathfrak{g}:\left[1,2\right]\to \mathbb{R}$ defined by $\mathfrak{g}\left(\mathsf{\zeta }\right)={\mathsf{\zeta }}^2$, and let $x=\frac{4}{3}$ and $\mathrm{y}=\frac{5}{3}.$ Then, we have ${}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\zeta }\right)=$${}^{\frac{5}{3}}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\zeta }\right)=\left(1+\widehat{\mathrm{q}}\right)\mathsf{\zeta }+\left(1-\widehat{\mathrm{q}}\right)\frac{5}{3}$, and $\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\right|$ is convex on $\left[1,2\right].$ Under these assumptions, we have

$$\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})=\frac{16}{9},$$

$$\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)={\left(\frac{4+5\widehat{\mathrm{q}}}{3{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)}^2$$

and

$$\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })=\frac{25}{9}.$$

From Definition 3, we get

$$\begin{array}{ccc}\hfill {\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda & =& {\int }_{\frac{4}{3}}^{\frac{5}{3}}{\lambda }^2{}^{\frac{5}{3}}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ & =& \frac{\left(1-\widehat{\mathrm{q}}\right)}{3}\sum _{n=0}^{\infty }{\widehat{\mathrm{q}}}^n{\left({\widehat{\mathrm{q}}}^n\frac{4}{3}+\left(1-{\widehat{\mathrm{q}}}^n\right)\frac{5}{3}\right)}^2\hfill \\ & =& \frac{\left(1-\widehat{\mathrm{q}}\right)}{27}\sum _{n=0}^{\infty }{\widehat{\mathrm{q}}}^n\left(25-10{\widehat{\mathrm{q}}}^n+{\widehat{\mathrm{q}}}^{2n}\right)\hfill \\ & =& \frac{1}{27}\left[25-\frac{10}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right].\hfill \end{array}$$

Hence, the left-hand side of the inequality (4) reduces to

$$\begin{array}{cc}\hfill & |\frac{1}{(\mathrm{y}-\mathsf{\varkappa })}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & =\left|\frac{1}{9}\left[25-\frac{10}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right]-\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\frac{16+25\widehat{\mathrm{q}}}{9}+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}{\left(\frac{4+5\widehat{\mathrm{q}}}{3{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)}^2\right]\right|.\hfill \end{array}$$

Since

$$\begin{array}{ccc}\hfill \left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|& =& \frac{8-2\widehat{\mathrm{q}}}{3},\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|=\frac{11+\widehat{\mathrm{q}}}{3}\hfill \\ \hfill \left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|& =& \frac{10}{3},\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|=\frac{9-\widehat{\mathrm{q}}}{3},\hfill \end{array}$$

(5)

we can write the right-hand side of the inequality (4) as

$$\begin{array}{cc}\hfill & \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })\left\{\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)|+|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\left|\right(\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+2\widehat{\mathrm{q}}\frac{(1+{\left[5\right]}_{\widehat{\mathrm{q}}}^2)}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\hfill \\ \hfill & -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3})-|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)|\left[{\mathfrak{c}}_{1\widehat{\mathrm{q}}}+{\mathfrak{c}}_{2\widehat{\mathrm{q}}}\right]-|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\left|[{\mathrm{d}}_{1\widehat{\mathrm{q}}}+{\mathrm{d}}_{2\widehat{\mathrm{q}}}]\right\}\hfill \\ \hfill & =\frac{\widehat{\mathrm{q}}}{3}\{\frac{8-2\widehat{\mathrm{q}}}{3}+\frac{11+\widehat{\mathrm{q}}}{3}(\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+2\widehat{\mathrm{q}}\frac{(1+{\left[5\right]}_{\widehat{\mathrm{q}}}^2)}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\hfill \\ \hfill & -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3})-\frac{10}{3}\left[{\mathfrak{c}}_{1\widehat{\mathrm{q}}}+{\mathfrak{c}}_{2\widehat{\mathrm{q}}}\right]-\frac{9-\widehat{\mathrm{q}}}{3}[{\mathrm{d}}_{1\widehat{\mathrm{q}}}+{\mathrm{d}}_{2\widehat{\mathrm{q}}}]\}.\hfill \end{array}$$

Then, by using the inequality (4), we can write

$$\begin{array}{cc}\hfill & \left|\frac{1}{9}\left[25-\frac{10}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right]-\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\frac{41}{9}+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}{\left(\frac{4+5\widehat{\mathrm{q}}}{3{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)}^2\right]\right|\hfill \\ \hfill & \le \frac{\widehat{\mathrm{q}}}{3}\{\frac{8-2\widehat{\mathrm{q}}}{3}+\frac{11+\widehat{\mathrm{q}}}{3}(\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+2\widehat{\mathrm{q}}\frac{(1+{\left[5\right]}_{\widehat{\mathrm{q}}}^2)}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\hfill \\ \hfill & -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3})-\frac{10}{3}\left[{\mathfrak{c}}_{1\widehat{\mathrm{q}}}+{\mathfrak{c}}_{2\widehat{\mathrm{q}}}\right]-\frac{9-\widehat{\mathrm{q}}}{3}[{\mathrm{d}}_{1\widehat{\mathrm{q}}}+{\mathrm{d}}_{2\widehat{\mathrm{q}}}]\}.\hfill \end{array}$$

(6)

One can see the validity of the inequality (6) in Figure 1.

Remark 3.

If we choose $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ in Theorem 2, we get Theorem 4 of [31].

Corollary 1.

With the assumptions of Theorem 2 with $\mathsf{\varkappa }={\rho }_1$, $\mathrm{y}={\rho }_2$, and $\widehat{\mathrm{q}}\to 1^{-}$, we have (see [35]):

$$\begin{array}{cc}\hfill & |\frac{1}{{\rho }_2-{\rho }_1}{\int }_{{\rho }_1}^{{\rho }_2}\mathfrak{g}\left(\lambda \right)d\lambda -\frac{1}{6}\left[\mathfrak{g}\left({\rho }_1\right)+4\mathfrak{g}\left(\frac{{\rho }_1+{\rho }_2}{2}\right)+\mathfrak{g}\left({\rho }_2\right)\right]|\hfill \\ \hfill & \le \frac{5({\rho }_2-{\rho }_1)}{36}\left(\frac{\left|{\mathfrak{g}}^{{}^{\prime }}\left({\rho }_1\right)\right|+\left|{\mathfrak{g}}^{{}^{\prime }}\left({\rho }_2\right)\right|}{2}\right).\hfill \end{array}$$

(7)

Theorem 3.

Let the assumptions of Lemma 3 hold. If ${\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\right|}^{{\wp }_{\mathfrak{1}}}$ is convex on $[{\rho }_1,{\rho }_2]$ for some fixed ${\wp }_{\mathfrak{1}}>1$, then the following inequality holds:

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left(\frac{{\widehat{\mathrm{q}}}^{2{\wp }_2}{\left[4\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\left[6\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}\right)}^{\frac{1}{{\wp }_2}}\times (\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|}^{{\wp }_1}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|}^{{\wp }_1}\hfill \\ \hfill & -\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|}^{{\wp }_1}-\frac{{\widehat{\mathrm{q}}}^2+2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left(\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\widehat{\mathrm{q}}}^{5{\wp }_2}-{\widehat{\mathrm{q}}}^{{\wp }_2}{\left[4\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\left[6\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}\right)}^{\frac{1}{{\wp }_2}}\times (\frac{\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|}^{{\wp }_1}+\frac{\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|}^{{\wp }_1}\hfill \\ \hfill & -\frac{{\widehat{\mathrm{q}}}^2+2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|}^{{\wp }_1}-\frac{{\widehat{\mathrm{q}}}^3+{\widehat{\mathrm{q}}}^2-\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}},\hfill \end{array}$$

where $0<\widehat{\mathrm{q}}<1$ and $\frac{1}{{\wp }_1}+\frac{1}{{\wp }_2}=1$.

Proof. 

Applying the well-known Hölder inequality for the ${\widehat{\mathrm{q}}}^{{\rho }_2}$-integral on the right-hand side of Lemma 3, we have

$$\begin{array}{cc}\hfill & |\frac{1}{(\mathrm{y}-\mathsf{\varkappa })}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_2}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \times {\left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_2}}\hfill \\ \hfill & \times {\left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_1}}.\hfill \end{array}$$

By using the convexity of $|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}{\mathfrak{g}|}^{{\wp }_1}$, we obtain:

$$\begin{array}{cc}\hfill & |\frac{1}{(\mathrm{y}-\mathsf{\varkappa })}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_2}}\times \left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda -{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\lambda d_{\widehat{\mathrm{q}}}\lambda |{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}-{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}(1-\lambda )d_{\widehat{\mathrm{q}}}\lambda |{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_2}}\times \left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda -{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1\lambda d_{\widehat{\mathrm{q}}}\lambda |{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}-{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1(1-\lambda )d_{\widehat{\mathrm{q}}}\lambda |{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}}.\hfill \end{array}$$

To calculate the integrals,

$$\begin{array}{cc}\hfill {\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda & =\frac{(1-\widehat{\mathrm{q}})}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\sum _{n=0}^{\infty }{\widehat{\mathrm{q}}}^n|\frac{{\widehat{\mathrm{q}}}^n}{{\left[2\right]}_{\widehat{\mathrm{q}}}}-\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}\hfill \\ \hfill & \le \frac{(1-\widehat{\mathrm{q}})}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\sum _{n=0}^{\infty }{\widehat{\mathrm{q}}}^n|\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}-\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}\hfill \\ \hfill & =\frac{{\widehat{\mathrm{q}}}^{2{\wp }_2}{\left[4\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\left[6\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}.\hfill \end{array}$$

Similarly,

$$\begin{array}{c}\hfill {\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda =\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\widehat{\mathrm{q}}}^{5{\wp }_2}-{\widehat{\mathrm{q}}}^{{\wp }_2}{\left[4\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\left[6\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}.\end{array}$$

Here, we also have

$$\begin{array}{c}\hfill {\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\lambda d_{\widehat{\mathrm{q}}}\lambda =\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3},\end{array}$$

$$\begin{array}{c}\hfill {\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}(1-\lambda )d_{\widehat{\mathrm{q}}}\lambda =\frac{{\widehat{\mathrm{q}}}^2+2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3},\end{array}$$

$$\begin{array}{c}\hfill {\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1\lambda d_{\widehat{\mathrm{q}}}\lambda =\frac{{\widehat{\mathrm{q}}}^2+2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3},\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1(1-\lambda )d_{\widehat{\mathrm{q}}}\lambda =\frac{{\widehat{\mathrm{q}}}^3+{\widehat{\mathrm{q}}}^2-\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}.\end{array}$$

By substituting these integrals, we get the required results. □

Remark 4.

If we put $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ into Theorem 3, then we get Theorem 5 of [31].

Theorem 4.

Let the assumptions of Lemma 3 hold. If ${\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\right|}^{{\wp }_{\mathfrak{1}}}$ is convex on $[{\rho }_1,{\rho }_2]$ for some fixed ${\wp }_{\mathfrak{1}}\ge 1$, then the following inequality holds:

$$\begin{array}{cc}\hfill & |\frac{1}{(\mathrm{y}-\mathsf{\varkappa })}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left(\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & \times (\frac{{\widehat{\mathrm{q}}}^3{\left[3\right]}_{\widehat{\mathrm{q}}}-\widehat{\mathrm{q}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\left\{{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|}^{{\wp }_1}+{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|}^{{\wp }_1}\right\}\hfill \\ \hfill & -{\mathfrak{c}}_{1\widehat{\mathrm{q}}}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|}^{{\wp }_1}-{\mathrm{d}}_{1\widehat{\mathrm{q}}}{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +{\left(\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+2\widehat{\mathrm{q}}\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}^2}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}\right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & \times (-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+2\widehat{\mathrm{q}}\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}^2}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[6\right]}_{\widehat{\mathrm{q}}}^2}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\left\{{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|}^{{\wp }_1}+{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|}^{{\wp }_1}\right\}\hfill \\ \hfill & -{\mathfrak{c}}_{2\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}-{\mathrm{d}}_{2\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}},\hfill \end{array}$$

where $0<\widehat{\mathrm{q}}<1,{\mathfrak{c}}_{1\widehat{\mathrm{q}}},{\mathfrak{c}}_{2\widehat{\mathrm{q}}},{\mathrm{d}}_{1\widehat{\mathrm{q}}}$, and ${\mathrm{d}}_{2\widehat{\mathrm{q}}}$ are defined in 2.

Proof. 

By applying the power mean inequality on the right-hand side of Lemma 3, we have

$$\begin{array}{cc}\hfill & |\frac{1}{(\mathrm{y}-\mathsf{\varkappa })}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & \times {\left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}-(1-\lambda )\mathsf{\varkappa }))\right|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & \times {\left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\right|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_1}}.\hfill \end{array}$$

Now, applying the convexity of ${|}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}{\mathfrak{g}|}^{{\wp }_1}$, we have

$$\begin{array}{cc}\hfill & |\frac{1}{(\mathrm{y}-\mathsf{\varkappa })}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}\left[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+{\widehat{\mathrm{q}}}^2{\left[4\right]}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}\mathsf{\varkappa }}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)+\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & \left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\right|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \left\{{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|}^{{\wp }_1}+{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|}^{{\wp }_1}\right\}\hfill \\ \hfill & -\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}\right|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda -|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}(1-\lambda )|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda {)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill +& \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & \times \left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1\right|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \left\{{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|}^{{\wp }_1}+|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right){|}^{{\wp }_1}\right\}\hfill \\ \hfill & -\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1\lambda \right|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda -|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|(1-\lambda )d_{\widehat{\mathrm{q}}}\lambda {)}^{\frac{1}{{\wp }_1}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_0^{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \right)}^{1-\frac{1}{{\wp }_1}}\times ({\mathfrak{c}}_{3\widehat{\mathrm{q}}}\left\{\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right){|}^{{\wp }_1}+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right){|}^{{\wp }_1}\right\}\hfill \\ \hfill & -{\mathfrak{c}}_{1\widehat{\mathrm{q}}}\left(\widehat{\mathrm{q}}\right)|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda -{\mathrm{d}}_{1\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda {)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_{\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[5\right]}_{\widehat{\mathrm{q}}}}{{\left[6\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda \right)}^{1-\frac{1}{{\wp }_1}}\times ({\mathrm{d}}_{3\widehat{\mathrm{q}}}\left\{|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right){|}^{{\wp }_1}+|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right){|}^{{\wp }_1}\right\}\hfill \\ \hfill & -{\mathfrak{c}}_{2\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda -{\mathrm{d}}_{2\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda {)}^{\frac{1}{{\wp }_1}}.\hfill \end{array}$$

By putting in the values of the integrals, we get required results. □

Remark 5.

If we put $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ into Theorem 4, we get Theorem 5 of [31].

Corollary 2.

With the suppositions of Theorem 4 with $\mathsf{\varkappa }={\rho }_1$, $\mathrm{y}={\rho }_2$, and $\widehat{\mathrm{q}}\to 1^{-}$, we have (see [35]):

$$\begin{array}{cc}\hfill & |\frac{1}{{\rho }_2-{\rho }_1}{\int }_{{\rho }_1}^{{\rho }_2}\mathfrak{g}\left(\lambda \right)d\lambda -\frac{1}{6}\left[\mathfrak{g}\left({\rho }_1\right)+4\mathfrak{g}\left(\frac{{\rho }_1+{\rho }_2}{2}\right)+\mathfrak{g}\left({\rho }_2\right)\right]|\hfill \\ \hfill & \le ({\rho }_2-{\rho }_1)\frac{1}{{\left(1296\right)}^{\frac{1}{{\wp }_1}}}{\left(\frac{5}{72}\right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & \times \left\{{\left(61{|{\mathfrak{g}}^{{}^{\prime }}\left({\rho }_1\right)|}^{{\wp }_1}+29{|{\mathfrak{g}}^{{}^{\prime }}\left({\rho }_2\right)|}^{{\wp }_1}\right)}^{\frac{1}{{\wp }_1}}+{\left(61{|{\mathfrak{g}}^{{}^{\prime }}\left({\rho }_1\right)|}^{{\wp }_1}+29{|{\mathfrak{g}}^{{}^{\prime }}\left({\rho }_2\right)|}^{{\wp }_1}\right)}^{\frac{1}{{\wp }_1}}\right\}.\hfill \end{array}$$

(8)

3.2. Simpson–Mercer $3/8$ Formula-Type Inequalities

Theorem 5.

With the suppositions of Theorem 2, we have

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\left]\right|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })\left[\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)|+|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\left|\right(-\frac{\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2}\frac{{\left[8\right]}_{\widehat{\mathrm{q}}}-{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\hfill \\ \hfill & -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[7\right]}_{\widehat{\mathrm{q}}}-{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{2\widehat{\mathrm{q}}\left[{\left[8\right]}_{\widehat{\mathrm{q}}}^2+{\left[2\right]}_{\widehat{\mathrm{q}}}^2(1+{\left[7\right]}_{\widehat{\mathrm{q}}}^2)\right]}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3{\left[8\right]}_{\widehat{\mathrm{q}}}^2})\hfill \\ \hfill & -\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|[{\mathfrak{c}}_{3\widehat{\mathrm{q}}}+{\mathfrak{c}}_{4\widehat{\mathrm{q}}}+{\mathfrak{c}}_{5\widehat{\mathrm{q}}}]-\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|[{\mathrm{d}}_{3\widehat{\mathrm{q}}}+{\mathrm{d}}_{4\widehat{\mathrm{q}}}+{\mathrm{d}}_{5\widehat{\mathrm{q}}}]]\hfill \end{array}$$

(9)

where

$${\omega }_{3\widehat{\mathrm{q}}}={\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}^3,$$

$$\begin{array}{cc}\hfill {\mathfrak{c}}_{3\widehat{\mathrm{q}}}& ={\int }_0^{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}\lambda |\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =\frac{{\left[8\right]}_{\widehat{\mathrm{q}}}^3{\left[2\right]}_{\widehat{\mathrm{q}}}-{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}^2+2{\widehat{\mathrm{q}}}^2{\left[3\right]}_{\widehat{\mathrm{q}}}^3}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^4{\left[8\right]}_{\widehat{\mathrm{q}}}^3},\hfill \\ \hfill {\mathfrak{c}}_{4\widehat{\mathrm{q}}}& =|\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|(1-\lambda )d_{\widehat{\mathrm{q}}}\lambda =\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}^2(1+{\left[2\right]}_{\widehat{\mathrm{q}}}^3)-{\left[3\right]}_{\widehat{\mathrm{q}}}^2(1+{\left[2\right]}_{\widehat{\mathrm{q}}}^2)}{{\left[2\right]}_{\widehat{\mathrm{q}}}^2{\left[3\right]}_{\widehat{\mathrm{q}}}^4}+\frac{2{\widehat{\mathrm{q}}}^2}{{\left[2\right]}_{\widehat{\mathrm{q}}}^4{\left[3\right]}_{\widehat{\mathrm{q}}}},\hfill \\ \hfill {\mathfrak{c}}_{5\widehat{\mathrm{q}}}& ={\int }_{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}\lambda |\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =\frac{2{\widehat{\mathrm{q}}}^2{\left[7\right]}_{\widehat{\mathrm{q}}}^3}{{\omega }_{3\widehat{\mathrm{q}}}}+\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}({\left[2\right]}_{\widehat{\mathrm{q}}}^3+{\left[3\right]}_{\widehat{\mathrm{q}}}^3)-({\left[2\right]}_{\widehat{\mathrm{q}}}^3+{\left[3\right]}_{\widehat{\mathrm{q}}}^3){\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^4{\left[8\right]}_{\widehat{\mathrm{q}}}},\hfill \\ \hfill {\mathrm{d}}_{3\widehat{\mathrm{q}}}& ={\int }_{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}(1-\lambda )|\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =2\frac{\widehat{\mathrm{q}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}-{\widehat{\mathrm{q}}}^2}{{\omega }_{3\widehat{\mathrm{q}}}}+\frac{1-{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2},\hfill \\ \hfill {\mathrm{d}}_{4\widehat{\mathrm{q}}}& ={\int }_{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^1\lambda |\lambda -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =\frac{2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}-\widehat{\mathrm{q}}\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2}-{\mathfrak{c}}_{4\widehat{\mathrm{q}}},\hfill \\ \hfill {\mathrm{d}}_{5\widehat{\mathrm{q}}}& ={\int }_{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^1(1-\lambda )|\lambda -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|d_{\widehat{\mathrm{q}}}\lambda =2\frac{\widehat{\mathrm{q}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[7\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}-{\widehat{\mathrm{q}}}^2{\left[7\right]}_{\widehat{\mathrm{q}}}^3}{{\omega }_{3\widehat{\mathrm{q}}}}+\frac{{\widehat{\mathrm{q}}}^2}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}}-\frac{\widehat{\mathrm{q}}{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}\hfill \\ \hfill & +\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}({\left[3\right]}_{\widehat{\mathrm{q}}}^2-{\left[2\right]}_{\widehat{\mathrm{q}}}^2)}{{\left[3\right]}_{\widehat{\mathrm{q}}}^4}-\frac{\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}^2{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}},\hfill \end{array}$$

and $0<\widehat{\mathrm{q}}<1$.

Proof. 

Applying the modulus to Lemma 4, we attain

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\left]\right|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\int }_0^{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\right|d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\int }_{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\right|d_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\int }_{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}|\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\right|d_{\widehat{\mathrm{q}}}\lambda .\hfill \end{array}$$

It is now sufficient to use similar steps to those in the results of Theorem 5 to find the desired outcomes. □

We give the following example to show the validity of the results given in Theorem 5.

Example 2.

Let $x,$$\mathrm{y}$, and the function $\mathfrak{g}$ be as in Example 1. Then, $\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\right|$ is convex, and we get

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\left]\right|\hfill \end{array}$$

$$\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})=\frac{16}{9},$$

$$\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)={\left(\frac{5{\left[3\right]}_{\widehat{\mathrm{q}}}-1}{3{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)}^2,$$

$$\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)={\left(\frac{4{\left[3\right]}_{\widehat{\mathrm{q}}}+{\widehat{\mathrm{q}}}^2}{3{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)}^2,$$

$$\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })=\frac{25}{9},$$

and

$${\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda =\frac{1}{27}\left[25-\frac{10}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right].$$

Therefore, we have the equality from (9),

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\left]\right|\hfill \\ \hfill & =|\frac{1}{9}\left[25-\frac{10}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right]-\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}\left[\frac{16+25\widehat{\mathrm{q}}}{9}+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left(\frac{5{\left[3\right]}_{\widehat{\mathrm{q}}}-1}{3{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)}^2+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left(\frac{4{\left[3\right]}_{\widehat{\mathrm{q}}}+{\widehat{\mathrm{q}}}^2}{3{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)}^2\right]|\hfill \end{array}$$

By using the equalities (5), we can write the right-hand side of the inequality (9) as

$$\begin{array}{cc}\hfill & \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })\left[\right|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)|+|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\left|\right(-\frac{\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2}\frac{{\left[8\right]}_{\widehat{\mathrm{q}}}-{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\hfill \\ \hfill & -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[7\right]}_{\widehat{\mathrm{q}}}-{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{2\widehat{\mathrm{q}}\left[{\left[8\right]}_{\widehat{\mathrm{q}}}^2+{\left[2\right]}_{\widehat{\mathrm{q}}}^2(1+{\left[7\right]}_{\widehat{\mathrm{q}}}^2)\right]}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3{\left[8\right]}_{\widehat{\mathrm{q}}}^2})\hfill \\ \hfill & -\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right)\right|[{\mathfrak{c}}_{3\widehat{\mathrm{q}}}+{\mathfrak{c}}_{4\widehat{\mathrm{q}}}+{\mathfrak{c}}_{5\widehat{\mathrm{q}}}]-\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right)\right|[{\mathrm{d}}_{3\widehat{\mathrm{q}}}+{\mathrm{d}}_{4\widehat{\mathrm{q}}}+{\mathrm{d}}_{5\widehat{\mathrm{q}}}]]\hfill \\ \hfill & =\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })[\frac{8-2\widehat{\mathrm{q}}}{3}+\frac{11+\widehat{\mathrm{q}}}{3}|(-\frac{\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2}\frac{{\left[8\right]}_{\widehat{\mathrm{q}}}-{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\hfill \\ \hfill & -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[7\right]}_{\widehat{\mathrm{q}}}-{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{2\widehat{\mathrm{q}}\left[{\left[8\right]}_{\widehat{\mathrm{q}}}^2+{\left[2\right]}_{\widehat{\mathrm{q}}}^2(1+{\left[7\right]}_{\widehat{\mathrm{q}}}^2)\right]}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3{\left[8\right]}_{\widehat{\mathrm{q}}}^2})\hfill \\ \hfill & -\frac{10}{3}[{\mathfrak{c}}_{3\widehat{\mathrm{q}}}+{\mathfrak{c}}_{4\widehat{\mathrm{q}}}+{\mathfrak{c}}_{5\widehat{\mathrm{q}}}]-\frac{9-\widehat{\mathrm{q}}}{3}[{\mathrm{d}}_{3\widehat{\mathrm{q}}}+{\mathrm{d}}_{4\widehat{\mathrm{q}}}+{\mathrm{d}}_{5\widehat{\mathrm{q}}}]].\hfill \end{array}$$

Finally, one can write the inequality (9) as follows and can see the validity of this inequality in Figure 2.

$$\begin{array}{cc}\hfill & |\frac{1}{9}\left[25-\frac{10}{{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right]-\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}\left[\frac{16+25\widehat{\mathrm{q}}}{9}+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left(\frac{5{\left[3\right]}_{\widehat{\mathrm{q}}}-1}{3{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)}^2+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}{\left(\frac{4{\left[3\right]}_{\widehat{\mathrm{q}}}+{\widehat{\mathrm{q}}}^2}{3{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)}^2\right]|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa })[\frac{8-2\widehat{\mathrm{q}}}{3}+\frac{11+\widehat{\mathrm{q}}}{3}|(-\frac{\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2}\frac{{\left[8\right]}_{\widehat{\mathrm{q}}}-{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\hfill \\ \hfill & -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[7\right]}_{\widehat{\mathrm{q}}}-{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}+\frac{2\widehat{\mathrm{q}}\left[{\left[8\right]}_{\widehat{\mathrm{q}}}^2+{\left[2\right]}_{\widehat{\mathrm{q}}}^2(1+{\left[7\right]}_{\widehat{\mathrm{q}}}^2)\right]}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3{\left[8\right]}_{\widehat{\mathrm{q}}}^2})\hfill \\ \hfill & -\frac{10}{3}[{\mathfrak{c}}_{3\widehat{\mathrm{q}}}+{\mathfrak{c}}_{4\widehat{\mathrm{q}}}+{\mathfrak{c}}_{5\widehat{\mathrm{q}}}]-\frac{9-\widehat{\mathrm{q}}}{3}[{\mathrm{d}}_{3\widehat{\mathrm{q}}}+{\mathrm{d}}_{4\widehat{\mathrm{q}}}+{\mathrm{d}}_{5\widehat{\mathrm{q}}}]].\hfill \end{array}$$

Remark 6.

If we choose $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ in Theorem 5, we get Theorem 6 of [31].

Theorem 6.

With the suppositions of Theorem 3, we have following inequality:

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\left]\right|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left(\frac{{\widehat{\mathrm{q}}}^{3{\wp }_2}{\left[5\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\left[8\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}\right)}^{\frac{1}{{\wp }_2}}\hfill \\ \hfill & \times {\left(\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|\right]-\frac{|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}-1}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}\right)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left(\frac{{\widehat{\mathrm{q}}}^{{\wp }_2}{\left[2\right]}_{\widehat{\mathrm{q}}}-{\widehat{\mathrm{q}}}^{2{\wp }_2}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}{\left[2\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}\right)}^{\frac{1}{{\wp }_2}}\hfill \\ \hfill & \times (\frac{\widehat{\mathrm{q}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|\right]\hfill \\ \hfill & -\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}^2-1}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}-\frac{{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}\widehat{\mathrm{q}}-{\left[2\right]}_{\widehat{\mathrm{q}}}^2+1}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left(\frac{{\widehat{\mathrm{q}}}^{7{\wp }_2}}{{\left[8\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}}-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{({\left[7\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}-{\left[8\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}})}^{{\wp }_2}}{{\left[8\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2}{\left[3\right]}_{\widehat{\mathrm{q}}}^{{\wp }_2+1}}\right)}^{\frac{1}{{\wp }_2}}\hfill \\ \hfill & \times (\frac{{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|\right]-\frac{{\left[3\right]}_{\widehat{\mathrm{q}}}^2-{\left[2\right]}_{\widehat{\mathrm{q}}}^2}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^2}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}\hfill \\ \hfill & -\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\widehat{\mathrm{q}}}^2+{\left[2\right]}_{\widehat{\mathrm{q}}}^2-{\left[3\right]}_{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}},\hfill \\ \hfill \end{array}$$

where $\frac{1}{{\wp }_1}+\frac{1}{{\wp }_2}=1$.

Proof. 

By applying Hölder’s inequality in Lemma 4, we attain

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_{\mathfrak{1}}+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_{\mathfrak{1}}+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\widehat{\mathrm{q}}\mathfrak{g}({\rho }_{\mathfrak{1}}+{\rho }_2-\mathsf{\varkappa })\left]\right|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_0^{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_2}}{\left({\int }_0^{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}|\lambda -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_2}}{\left({\int }_{\frac{1}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_{\mathfrak{1}}+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa })){|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +\widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left({\int }_{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^1|\lambda -\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}{|}^{{\wp }_2}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_2}}{\left({\int }_{\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}}}^1{\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}({\rho }_1+{\rho }_2-(\lambda \mathrm{y}+(1-\lambda )\mathsf{\varkappa }))\right|}^{{\wp }_1}{d}_{\widehat{\mathrm{q}}}\lambda \right)}^{\frac{1}{{\wp }_1}}.\hfill \end{array}$$

By applying a similar method to that used in the results of Theorem 3, we find the desired outcomes. □

Remark 7.

If we put $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ into Theorem 6, we get Theorem 8 of [31].

Theorem 7.

With the suppositions of Theorem 4, we have the inequality

$$\begin{array}{cc}\hfill & |\frac{1}{\mathrm{y}-\mathsf{\varkappa }}{\int }_{{\rho }_1+{\rho }_2-\mathrm{y}}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}\mathfrak{g}\left(\lambda \right){}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{d}_{\widehat{\mathrm{q}}}\lambda \hfill \\ \hfill & -\frac{1}{{\left[8\right]}_{\widehat{\mathrm{q}}}}[\mathfrak{g}({\rho }_1+{\rho }_2-\mathrm{y})+\frac{{\widehat{\mathrm{q}}}^3{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{\mathrm{y}+\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}\mathsf{\varkappa }}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)+\frac{{\widehat{\mathrm{q}}}^2{\left[6\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}}\mathfrak{g}\left({\rho }_1+{\rho }_2-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}\mathrm{y}+\mathsf{\varkappa }{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}}\right)\hfill \\ \hfill & +\widehat{\mathrm{q}}\mathfrak{g}({\rho }_1+{\rho }_2-\mathsf{\varkappa })\left]\right|\hfill \\ \hfill & \le \widehat{\mathrm{q}}(\mathrm{y}-\mathsf{\varkappa }){\left(\frac{2\widehat{\mathrm{q}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{{\left[8\right]}_{\widehat{\mathrm{q}}}-{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}\right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & (\frac{2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}^2}+\frac{{\left[8\right]}_{\widehat{\mathrm{q}}}-{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}\left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|\right]\hfill \\ \hfill & -{\mathfrak{c}}_{3\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}-{\mathrm{d}}_3\left(\widehat{\mathrm{q}}\right)|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +{\left(\frac{2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}+\frac{\widehat{\mathrm{q}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{2\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}+{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}\right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & {\left(\frac{2\widehat{\mathrm{q}}}{{\left[2\right]}_{\widehat{\mathrm{q}}}^3}-\frac{\widehat{\mathrm{q}}{\left[2\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2}\left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_1\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|\right]-{\mathfrak{c}}_{4\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}-{\mathrm{d}}_{4\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}\right)}^{\frac{1}{{\wp }_1}}\hfill \\ \hfill & +{\left(2\frac{\widehat{\mathrm{q}}{\left[7\right]}_{\widehat{\mathrm{q}}}^2}{{\left[8\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}+\frac{{\left[3\right]}_{\widehat{\mathrm{q}}}^2+{\left[2\right]}_{\widehat{\mathrm{q}}}^2}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[2\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}({\left[3\right]}_{\widehat{\mathrm{q}}}+{\left[2\right]}_{\widehat{\mathrm{q}}})}{{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}\right)}^{1-\frac{1}{{\wp }_1}}\hfill \\ \hfill & (\frac{2\widehat{\mathrm{q}}{\left[7\right]}_{\widehat{\mathrm{q}}}^2}{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}^2}+\frac{1}{{\left[2\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[7\right]}_{\widehat{\mathrm{q}}}}{{\left[8\right]}_{\widehat{\mathrm{q}}}}-\frac{{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[3\right]}_{\widehat{\mathrm{q}}}{\left[7\right]}_{\widehat{\mathrm{q}}}-{\left[2\right]}_{\widehat{\mathrm{q}}}{\left[8\right]}_{\widehat{\mathrm{q}}}}{{\left[3\right]}_{\widehat{\mathrm{q}}}^2{\left[8\right]}_{\widehat{\mathrm{q}}}}\left[\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_{\mathfrak{1}}\right)\right|+\left|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left({\rho }_2\right)\right|\right]\hfill \\ \hfill & -{\mathfrak{c}}_{5\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathrm{y}\right){|}^{{\wp }_1}-{\mathrm{d}}_{5\widehat{\mathrm{q}}}|{}^{{\rho }_1+{\rho }_2-\mathsf{\varkappa }}{D}_{\widehat{\mathrm{q}}}\mathfrak{g}\left(\mathsf{\varkappa }\right){|}^{{\wp }_1}{)}^{\frac{1}{{\wp }_1}},\hfill \end{array}$$

${\mathfrak{c}}_{3\widehat{\mathrm{q}}},{\mathfrak{c}}_{4\widehat{\mathrm{q}}},{\mathfrak{c}}_{5\widehat{\mathrm{q}}},{\mathrm{d}}_{3\widehat{\mathrm{q}}},{\mathrm{d}}_{4\widehat{\mathrm{q}}}$, and ${\mathrm{d}}_{5\widehat{\mathrm{q}}}$ are defined in Theorem 5.

Proof. 

The proof of the theorem is analogous to the proof of Theorem 4. □

Remark 8.

If we put $\mathsf{\varkappa }={\rho }_1$ and $\mathrm{y}={\rho }_2$ into Theorem 7, we get Theorem 9 of [31].

4. Conclusions

In this article, we developed new analogs of Simpson–Mercer-like identities. Thus, by employing quantum integration and the Jensen–Mercer inequality for convex functions, we computed new quantum bounds. This novel framework is a convolution of the Simpon–Mercer concept and the estimation of the $\widehat{\mathrm{q}}$-definite integral. Some examples are also provided for the main inequalities. By employing the quantum Hölder and power mean integral inequalities, we analyzed new quantum inequalities that are in connection with previously published results. Little work has been done on quantum Mercer inequalities to date, so it is interesting to extend these findings to other general inequalities and convexities. One of the important problems is to check the validity of such results for their coordinate convexity. We presume that our newly announced concept will be the focus of much research in this fascinating field of quantum inequalities.