Some Simpson-like Inequalities Involving the (s,m)-Preinvexity

Abstract

In this article, closed Newton–Cotes-type symmetrical inequalities involving four-point functions whose second derivatives are $(s,m)$-preinvex in the second sense are established. Some applications to quadrature formulas as well as inequalities involving special means are provided.

1. Introduction

One of the important concepts of real analysis and mathematical programming is convexity. Due to its various applications in various fields such as applied mathematics, engineering sciences and other fields, this notion has been extended and generalized in several directions and in various ways.

In [1], Meftah introduced the class of $\left(s,m\right)$-preinvex functions.

Definition 1

([1]). Function $\mathcal{C}:K\subset \left[0,b^{*}\right]\to \mathbb{R}$ is said to be $\left(s,m\right)$-preinvex with respect to η, where $\eta \left(.,.\right):K\times K\to \mathbb{R},b^{*}>0$ and $s,m\in \left(0,1\right]$, if

$$\mathcal{C}\left(l+j\eta \left(e,l\right)\right)\le {\left(1-j\right)}^s\mathcal{C}\left(l\right)+m{j}^s\mathcal{C}\left(\frac{e}{m}\right)$$

holds for all $l,e\in K$, and $j\in [0,1]$.

The previous class of function includes several classes depending on the values of $s,m$ and $\eta \left(.,.\right)$. Among these classes, we have convex functions [2], m-convex functions [3], s-convex functions in the second sense [4], (s, m)-convex functions in the second sense [5], preinvex functions [6], m-preinvex functions [7] and s-preinvex functions in the second sense [8].

In numerical analysis, numerous quadrature rules have been established to approximate the integrals defined under the aforementioned convexity classes; see [9,10,11,12,13,14,15,16,17,18,19,20].

The following inequalities are well known in the literature as Simpson’s inequalities:

$$\left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+4\mathcal{C}\left(\frac{l+e}{2}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\le \frac{1}{2880}{∥{\mathcal{C}}^{\left(4\right)}∥}_{\infty }{\left(e-l\right)}^4,$$

and

$$\left|\frac{1}{8}\left(\mathcal{C}\left(l\right)+3\mathcal{C}\left(\frac{2l+e}{3}\right)+3\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\le \frac{1}{6480}{∥{\mathcal{C}}^{\left(4\right)}∥}_{\infty }{\left(e-l\right)}^4,$$

where f is a four times continuously differentiable function on $\left[l,e\right]$, and ${∥{\mathcal{C}}^{\left(4\right)}∥}_{\infty }=\underset{x\in \left[l,e\right]}{sup}\left|{\mathcal{C}}^{\left(4\right)}\left(x\right)\right|$.

As the above inequalities are very popular in estimating errors of quadrature rules, many researchers have massively studied them, as well as similar inequalities; one can consult, for example, [21,22,23,24,25,26,27,28,29,30] and references therein.

In [31], Hua et al. offered the following Simpson-type inequalities:

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{6^{\frac{1}{q}}{\left(e-l\right)}^2}{324}\left({\left(\frac{\left(s-3\right)3^{2+s}+\left(7+s\right)2^{2+s}}{3^s\left(1+s\right)\left(2+s\right)\left(3+s\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+\frac{1}{3^s\left(2+s\right)\left(3+s\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q-c\frac{{\left(e-l\right)}^2}{45}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\frac{\left(s-1\right)2^{2+s}+5+s}{3^s\left(1+s\right)\left(2+s\right)\left(3+s\right)}\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q\right)-c\frac{11{\left(e-l\right)}^2}{270}\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\frac{1}{3^s\left(2+s\right)\left(3+s\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+\frac{\left(s-3\right)3^{2+s}+\left(7+s\right)2^{2+s}}{3^s\left(1+s\right)\left(2+s\right)\left(3+s\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q-c\frac{{\left(e-l\right)}^2}{45}\right)}^{\frac{1}{q}}\right)\hfill \end{array}$$

and

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{{\left(e-l\right)}^2}{54}{\left(\frac{1}{3^s\left(1+s\right)}\right)}^{\frac{1}{q}}{\left(B\left(\frac{2q-1}{q-1},\frac{2q-1}{q-1}\right)\right)}^{1-\frac{1}{q}}\hfill \\ & & \times \left({\left(\left(3^{1+s}-2^{1+s}\right){\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q-c\frac{7{\left(e-l\right)}^2{3}^s\left(1+s\right)}{54}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\left(2^{1+s}-1\right)\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q\right)-c\frac{13{\left(e-l\right)}^2{3}^s\left(1+s\right)}{54}\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+\left(3^{1+s}-2^{1+s}\right){\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q-c\frac{7{\left(e-l\right)}^2{3}^s\left(1+s\right)}{54}\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

In [32], Chiheb et al. established some Simpson-type inequalities for functions whose second derivatives are prequasi-invex, the results of which are based on the following identity:

Lemma 1

([32]). We let $\mathcal{C}:\left[l,l+\eta \left(e,l\right)\right]\to \mathbb{R}$ be a function such that ${\mathcal{C}}^{\prime }$ is absolutely continuous and ${\mathcal{C}}^{\prime \prime }$ is integrable on $\left[l,l+\eta \left(e,l\right)\right]$, then the following equality holds:

$$\begin{array}{ccc}\hfill F\left(e,l,\mathcal{C}\right)& =& \frac{{\eta }^2\left(e,l\right)}{54}\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left\{{\mathcal{C}}^{\prime \prime }\left(l+\frac{1-j}{3}\eta \left(e,l\right)\right)\right.\hfill \\ & & +\left.{\mathcal{C}}^{\prime \prime }\left(l+\frac{2-j}{3}\eta \left(e,l\right)\right)+{\mathcal{C}}^{\prime \prime }\left(l+\frac{3-j}{3}\eta \left(e,l\right)\right)\right\}dj,\hfill \end{array}$$

(1)

where

$$\begin{array}{ccc}\hfill F\left(e,l,\mathcal{C}\right)& =& \frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{3l+\eta \left(e,l\right)}{3}\right)+2\mathcal{C}\left(\frac{3l+2\eta \left(e,l\right)}{3}\right)+\mathcal{C}\left(l+\eta \left(e,l\right)\right)\right)\hfill \\ & & -\frac{1}{\eta \left(e,l\right)}\underset{l}{\stackrel{l+\eta \left(e,l\right)}{\int }}\mathcal{C}\left(u\right)du.\hfill \end{array}$$

(2)

In this paper, using the identity declared in [32], we establish some Simpson-type inequalities for twice differentiable $(s,m)$-preinvex functions. Some special cases are derived. Applications to quadrature formulas and inequalities involving means are provided.

2. Main Results

The following special functions as well as the algebraic inequality are useful to our study.

Definition 2

([33]). The beta function is defined for $Rel$>0 and $Ree$>0 as follows:

$$B\left(l,e\right)=\underset{0}{\stackrel{1}{\int }}{j}^{l-1}{\left(1-j\right)}^{e-1}dj.$$

Definition 3

([34]). The hypergeometric function is defined for $Rec>Ree>0$ and $\left|z\right|<1$ as follows:

$${}_2{F}_1\left(l,e,c;z\right)=\frac{1}{B\left(e,c-e\right)}\underset{0}{\stackrel{1}{\int }}{j}^{e-1}{\left(1-j\right)}^{c-e-1}{\left(1-zj\right)}^{-l}dj.$$

Lemma 2

([35] Discrete Power mean inequality). For any $l,e>0$ and $0\le \zeta \le 1$, we have

$$l^{\zeta }+e^{\zeta }\le 2^{1-\zeta }{\left(l+e\right)}^{\zeta }.$$

Theorem 1.

We let $\mathcal{C}:\left[l,l+\eta \left(e,l\right)\right]\subset \left(0,\infty \right)\to \mathbb{R}$ be a twice differentiable function such that ${\mathcal{C}}^{\prime \prime }$$\in L$$\left[l,l+\eta \left(e,l\right)\right]$. If $\left|{\mathcal{C}}^{\prime \prime }\right|$ is $\left(s,m\right)$-preinvex for some fixed $s,m\in \left(0,1\right]$, we have

$$\left|F\left(e,l,\mathcal{C}\right)\right|\le \frac{{\eta }^2\left(e,l\right)}{54}\left(\frac{\left(2+2^{s+3}+3^{s+2}\right)s+\left(6+3\times 2^{s+3}-3^{s+3}\right)}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)\left(\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|+m\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|\right),$$

where $F\left(e,l,\mathcal{C}\right)$ is defined as in (2).

Proof. 

From Lemma 1, properties of modulus and $\left(s,m\right)$-preinvexity of $\left|{\mathcal{C}}^{\prime \prime }\right|$ on $\left[l,l+\eta \left(e,l\right)\right]$, we have

$$\begin{array}{ccc}& & \left|F\left(e,l,\mathcal{C}\right)\right|\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|f^{\prime \prime }\left(l+\frac{1-j}{3}\eta \left(e,l\right)\right)\right|dj\right.\hfill \\ & & +\left.\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|f^{\prime \prime }\left(l+\frac{2-j}{3}\eta \left(e,l\right)\right)\right|dj+\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|f^{\prime \prime }\left(l+\frac{3-j}{3}\eta \left(e,l\right)\right)\right|dj\right)\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left({\left(\frac{2+j}{3}\right)}^s\left|f^{\prime \prime }\left(l\right)\right|+m{\left(\frac{1-j}{3}\right)}^s\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\right)dj\right.\hfill \\ & & +\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left({\left(\frac{1+j}{3}\right)}^s\left|f^{\prime \prime }\left(l\right)\right|+m{\left(\frac{2-j}{3}\right)}^s\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\right)dj\hfill \\ & & +\left.\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left({\left(\frac{j}{3}\right)}^s\left|f^{\prime \prime }\left(l\right)\right|+m{\left(\frac{3-j}{3}\right)}^s\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\right)dj\right)\hfill \\ & =& \frac{{\eta }^2\left(e,l\right)}{54}\left(\left|f^{\prime \prime }\left(l\right)\right|\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{2+j}{3}\right)}^{s}dj+m\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{1-j}{3}\right)}^{s}dj\right.\hfill \\ & & +\left|f^{\prime \prime }\left(l\right)\right|\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{1+j}{3}\right)}^{s}dj+m\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{2-j}{3}\right)}^{s}dj\hfill \\ & & +\left.\left|f^{\prime \prime }\left(l\right)\right|\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{j}{3}\right)}^{s}dj+m\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{3-j}{3}\right)}^{s}dj\right)\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{{\eta }^2\left(e,l\right)}{54}\left(\left|f^{\prime \prime }\left(l\right)\right|+m\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\right)\hfill \\ & & \times \left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{2+j}{3}\right)}^{s}dj+\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{1+j}{3}\right)}^{s}dj+\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{j}{3}\right)}^{s}dj\right)\hfill \\ & =& \frac{{\eta }^2\left(e,l\right)}{54}\left(\left|f^{\prime \prime }\left(l\right)\right|+m\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\right)\hfill \\ & & \times \left(\frac{\left(2^{s+2}+3^{s+2}\right)s+7\times 2^{s+2}-3^{s+3}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}+\frac{\left(2^{s+2}+1\right)s+5-2^{s+2}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}+\frac{s+1}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)\hfill \\ & =& \frac{{\eta }^2\left(e,l\right)}{54}\left(\frac{\left(2+2^{s+3}+3^{s+2}\right)s+\left(6+3\times 2^{s+3}-3^{s+3}\right)}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)\left(\left|f^{\prime \prime }\left(l\right)\right|+m\left|f^{\prime \prime }\left(\frac{e}{m}\right)\right|\right),\hfill \end{array}$$

where we used the facts that

$$\begin{array}{ccc}\hfill \underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{2+j}{3}\right)}^{s}dj& =& \underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{3-j}{3}\right)}^{s}dj\hfill \\ & =& \frac{\left(2^{s+2}+3^{s+2}\right)s+7\times 2^{s+2}-3^{s+3}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)},\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{1+j}{3}\right)}^{s}dj& =& \underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{2-j}{3}\right)}^{s}dj=\frac{\left(2^{s+2}+1\right)s+5-2^{s+2}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\hfill \end{array}$$

(4)

and

$$\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{j}{3}\right)}^{s}dj=\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{1-j}{3}\right)}^{s}dj=\frac{1}{3^s\left(s+2\right)\left(s+3\right)}.$$

(5)

The proof is finished. □

Corollary 1.

Taking $s=m=1$ and $\eta \left(e,l\right)=e-l$, Theorem 1 becomes

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{{\left(e-l\right)}^2}{216}\left(\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|+\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|\right).\hfill \end{array}$$

Theorem 2.

Under the assumptions of Theorem 1, if ${\left|{\mathcal{C}}^{\prime \prime }\right|}^q$ where $q>1$, is $\left(s,m\right)$-preinvex in the second sense for some fixed $s,m\in \left(0,1\right]$, with $\frac{1}{p}+\frac{1}{q}=1$, we have

$$\begin{array}{ccc}& & \left|F\left(l,e,\mathcal{C}\right)\right|\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}{\left(B\left(p+1,p+1\right)\right)}^{\frac{1}{p}}\left({\left(\frac{\left(3^{s+1}-2^{s+1}\right){\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q}{3^s\left(1+s\right)}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +\left.{\left(\frac{\left(2^{s+1}-1\right)\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}{3^s\left(1+s\right)}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\left(3^{s+1}-2^{s+1}\right){\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q}{3^s\left(1+s\right)}\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where $F\left(l,e,\mathcal{C}\right)$ is defined as in (1.2).

Proof. 

From Lemma 1, properties of modulus, Hölder’s inequality and $\left(s,m\right)$-preinvexity of ${\left|{\mathcal{C}}^{\prime \prime }\right|}^q$ on $\left[l,l+\eta \left(e,l\right)\right]$, we have

$$\begin{array}{ccc}& & \left|F\left(l,e,\mathcal{C}\right)\right|\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{1-j}{3}\eta \left(e,l\right)\right)\right|dj\right.\hfill \\ & & +\left.\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{2-j}{3}\eta \left(e,l\right)\right)\right|dj+\underset{0}{\stackrel{1}{\int }}t\left(1-t\right)\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{3-j}{3}\eta \left(e,l\right)\right)\right|dj\right)\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}{\left(\underset{0}{\stackrel{1}{\int }}{j}^p{\left(1-j\right)}^{p}dj\right)}^{\frac{1}{p}}\left({\left(\underset{0}{\stackrel{1}{\int }}\left({\left(\frac{2+j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left(\frac{1-j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)dj\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left({\int }_0^1\left({\left(\frac{1+j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left(\frac{2-j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)dj\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left({\int }_0^1\left({\left(\frac{j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left(\frac{3-j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)dj\right)}^{\frac{1}{q}}\right)\hfill \\ & =& \frac{{\eta }^2\left(e,l\right)}{54\times 3^{\frac{s}{q}}}{\left(\underset{0}{\stackrel{1}{\int }}{j}^p{\left(1-j\right)}^{p}dj\right)}^{\frac{1}{p}}\hfill \\ & & \times \left({\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{\left(2+j\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{\left(1-j\right)}^{s}dj\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{\left(1+j\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{\left(2-j\right)}^{s}dj\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{j}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{\left(3-j\right)}^{s}dj\right)}^{\frac{1}{q}}\right)\hfill \\ & =& \frac{{\eta }^2\left(e,l\right)}{54}{\left(B\left(p+1,p+1\right)\right)}^{\frac{1}{p}}\left({\left(\frac{\left(3^{s+1}-2^{s+1}\right){\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q}{3^s\left(1+s\right)}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +\left.{\left(\frac{\left(2^{s+1}-1\right)\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}{3^s\left(1+s\right)}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\left(3^{s+1}-2^{s+1}\right){\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q}{3^s\left(1+s\right)}\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

which completes the proof. □

Corollary 2.

Taking $s=m=1$ and $\eta \left(e,l\right)=e-l$, Theorem 2 becomes

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{{\left(e-l\right)}^2}{54}{\left(B\left(p+1,p+1\right)\right)}^{\frac{1}{p}}\left({\left(\frac{5{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{6}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +\left.{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{2}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+5{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{6}\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

Corollary 3.

In Corollary 2, using the discrete power mean inequality, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{{\left(e-l\right)}^2}{18}{\left(B\left(p+1,p+1\right)\right)}^{\frac{1}{p}}{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{2}\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Theorem 3.

Under the assumptions of Theorem 2, if ${\left|{\mathcal{C}}^{\prime \prime }\right|}^q$ is $\left(s,m\right)$-preinvex in the second sense for some fixed $s,m\in \left(0,1\right]$ and $q\ge 1$, we have

$$\begin{array}{ccc}& & \left|F\left(l,e,\mathcal{C}\right)\right|\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54\times 6^{1-\frac{1}{q}}}\left({\left(\frac{\left(2^{s+2}+3^{s+2}\right)s+7\times 2^{s+2}-3^{s+3}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+\frac{m}{3^s\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\frac{\left(2^{s+2}+1\right)s+5-2^{s+2}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)}^{\frac{1}{q}}{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\frac{1}{3^s\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{\left(2^{s+2}+3^{s+2}\right)s+7\times 2^{s+2}-3^{s+3}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where $F\left(l,e,\mathcal{C}\right)$ is defined as in (1.2).

Proof. 

From Lemma 1, properties of modulus, power mean inequality and $\left(s,m\right)$-preinvexity of ${\left|f^{\prime \prime }\right|}^q$ on $\left[l,l+\eta \left(e,l\right)\right]$, we obtain

$$\begin{array}{ccc}& & \left|F\left(l,e,\mathcal{C}\right)\right|\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{1-j}{3}\eta \left(e,l\right)\right)\right|dj\right.\hfill \\ & & +\left.\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{2-j}{3}\eta \left(e,l\right)\right)\right|dj+\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{3-j}{3}\eta \left(e,l\right)\right)\right|dj\right)\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}{\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)dt\right)}^{1-\frac{1}{q}}\left({\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{1-j}{3}\eta \left(e,l\right)\right)\right|}^{q}dj\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{2-j}{3}\eta \left(e,l\right)\right)\right|}^{q}dj\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{3-j}{3}\eta \left(e,l\right)\right)\right|}^{q}dj\right)}^{\frac{1}{q}}\right)\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}{\left(\frac{1}{6}\right)}^{1-\frac{1}{q}}\left({\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left({\left(\frac{2+j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left(\frac{1-j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)dj\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left({\left(\frac{1+j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left(\frac{2-j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)dj\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\underset{0}{\stackrel{1}{\int }}j\left(1-j\right)\left({\left(\frac{j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left(1-\frac{j}{3}\right)}^s{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)dj\right)}^{\frac{1}{q}}\right)\hfill \end{array}$$

$$\begin{array}{ccc}& =& \frac{{\eta }^2\left(e,l\right)}{54\times 6^{1-\frac{1}{q}}}\left({\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{2+j}{3}\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{1-j}{3}\right)}^{s}dj\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{1+j}{3}\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{2-j}{3}\right)}^{s}dj\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(\frac{j}{3}\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}j\left(1-j\right){\left(1-\frac{j}{3}\right)}^{s}dj\right)}^{\frac{1}{q}}\right)\hfill \\ & =& \frac{{\eta }^2\left(e,l\right)}{54\times 6^{1-\frac{1}{q}}}\left({\left(\frac{\left(2^{s+2}+3^{s+2}\right)s+7\times 2^{s+2}-3^{s+3}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+\frac{m}{3^s\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\frac{\left(2^{s+2}+1\right)s+5-2^{s+2}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)}^{\frac{1}{q}}{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\frac{1}{3^s\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{\left(2^{s+2}+3^{s+2}\right)s+7\times 2^{s+2}-3^{s+3}}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where we used (2.1)–(2.3). The proof is completed. □

Corollary 4.

Taking $s=m=1$ and $\eta \left(e,l\right)=e-l$, Theorem 3 becomes

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{{\left(e-l\right)}^2}{324}\left({\left(\frac{5{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{6}\right)}^{\frac{1}{q}}+{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{2}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +\left.{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+5{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{6}\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

Corollary 5.

In Corollary 4, using the discrete power mean inequality, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{{\left(e-l\right)}^2}{108}{\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{2}\right)}^{\frac{1}{q}}.\hfill \end{array}$$

Theorem 4.

Under the assumptions of Theorem 2, we have the following inequality:

$$\begin{array}{ccc}& & \left|F\left(l,e,\mathcal{C}\right)\right|\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left({\left(\frac{{}_2{F}_1\left(-s,1,q+2;\frac{1}{3}\right)}{q+1}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{B\left(q+1,s+1\right)}{3^s}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\frac{2^s{\times }_2{F}_1\left(-s,1,q+2;\frac{1}{2}\right)}{3^s\left(q+1\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{2^s{\times }_2{F}_1\left(-s,q+1,q+2;\frac{1}{2}\right)}{3^s\left(q+1\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\frac{1}{3^s\left(q+s+1\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{{}_2{F}_1\left(-s,q+1,q+2;\frac{1}{3}\right)}{q+1}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where $F\left(l,e,\mathcal{C}\right)$ is defined as in (1.2) and B and ${}_2{F}_1$ are beta and hypergeometric functions, respectively.

Proof. 

From Lemma 1, properties of modulus, Hölder’s inequality and $\left(s,m\right)$-preinvexity of ${\left|f^{\prime \prime }\right|}^q$ on $\left[a,a+\eta \left(b,a\right)\right]$, we have

$$\begin{array}{ccc}& & \left|F\left(l,e,\mathcal{C}\right)\right|\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}{\left(\underset{0}{\stackrel{1}{\int }}{\left(1-j\right)}^{p}dt\right)}^{\frac{1}{p}}\left({\left(\underset{0}{\stackrel{1}{\int }}{j}^q{\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{1-j}{3}\eta \left(e,l\right)\right)\right|}^{q}dj\right)}^{\frac{1}{q}}\right.\hfill \\ & & +\left.{\left(\underset{0}{\stackrel{1}{\int }}{j}^q{\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{2-j}{3}\eta \left(e,l\right)\right)\right|}^{q}dj\right)}^{\frac{1}{q}}+{\left(\underset{0}{\stackrel{1}{\int }}{j}^q{\left|{\mathcal{C}}^{\prime \prime }\left(l+\frac{3-j}{3}\eta \left(e,l\right)\right)\right|}^{q}dj\right)}^{\frac{1}{q}}\right)\hfill \\ & \le & \frac{{\eta }^2\left(e,l\right)}{54}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left({\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{2+j}{3}\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{1-j}{3}\right)}^{s}dj\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{1+j}{3}\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{2-j}{3}\right)}^{s}dj\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left({\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{j}{3}\right)}^{s}dj+m{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{3-j}{3}\right)}^{s}dj\right)}^{\frac{1}{q}}\right)\hfill \\ & =& \frac{{\eta }^2\left(e,l\right)}{54}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left({\left(\frac{{}_2{F}_1\left(-s,1,q+2;\frac{1}{3}\right)}{q+1}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{B\left(q+1,s+1\right)}{3^s}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\frac{2^s{\times }_2{F}_1\left(-s,1,q+2;\frac{1}{2}\right)}{3^s\left(q+1\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{2^s{\times }_2{F}_1\left(-s,q+1,q+2;\frac{1}{2}\right)}{3^s\left(q+1\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\frac{1}{3^s\left(q+s+1\right)}{\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+m\frac{{}_2{F}_1\left(-s,q+1,q+2;\frac{1}{3}\right)}{q+1}{\left|{\mathcal{C}}^{\prime \prime }\left(\frac{e}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where we used

$$\begin{array}{ccc}\hfill \underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{2+j}{3}\right)}^{s}dj& =& \frac{1}{q+1}{.}_2{F}_1\left(-s,1,q+2;\frac{1}{3}\right),\hfill \\ \hfill \underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{1-j}{3}\right)}^{s}dj& =& {\left(\frac{1}{3}\right)}^{s}B\left(q+1,s+1\right),\hfill \\ \hfill \underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{1+j}{3}\right)}^{s}dj& =& {\left(\frac{2}{3}\right)}^s\frac{1}{q+1}{.}_2{F}_1\left(-s,1,q+2;\frac{1}{2}\right),\hfill \\ \hfill \underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{2-t}{3}\right)}^{s}dj& =& {\left(\frac{2}{3}\right)}^s\frac{1}{q+1}{.}_2{F}_1\left(-s,q+1,q+2;\frac{1}{2}\right),\hfill \\ \hfill \underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{j}{3}\right)}^{s}dj& =& {\left(\frac{1}{3}\right)}^s\frac{1}{q+s+1},\hfill \\ \hfill \underset{0}{\stackrel{1}{\int }}{j}^q{\left(\frac{3-j}{3}\right)}^{s}dj& =& \frac{1}{q+1}{.}_2{F}_1\left(-s,q+1,q+2;\frac{1}{3}\right).\hfill \end{array}$$

The proof is completed. □

Corollary 6.

In Theorem 4, taking $s=m=1$ and $\eta \left(e,l\right)=e-l$, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(l\right)+2\mathcal{C}\left(\frac{2l+e}{3}\right)+2\mathcal{C}\left(\frac{l+2e}{3}\right)+\mathcal{C}\left(e\right)\right)-\frac{1}{e-l}\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \frac{{\left(e-l\right)}^2}{162}{\left(\frac{3}{p+1}\right)}^{\frac{1}{p}}\left({\left(\frac{\left(3q+5\right){\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +\left.{\left(\frac{\left(2q+3\right){\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+\left(q+3\right){\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}+{\left(\frac{\left(q+1\right){\left|{\mathcal{C}}^{\prime \prime }\left(l\right)\right|}^q+\left(2q+5\right){\left|{\mathcal{C}}^{\prime \prime }\left(e\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

3. Applications

We let ${\rm Y}$ be the partition of points $l=x_0<x_1<\cdots <x_n=a$ of interval $\left[l,e\right]$ and consider quadrature formula

$$\stackrel{e}{\underset{l}{\int }}\mathcal{C}\left(u\right)du=\lambda \left(\mathcal{C},{\rm Y}\right)+R\left(\mathcal{C},{\rm Y}\right),$$

where

$$\lambda \left(\mathcal{C},{\rm Y}\right)=\underset{i=0}{\sum ^{n-1}}\frac{x_{i+1}-x_i}{6}\left(\mathcal{C}\left(x_i\right)+2\mathcal{C}\left(\frac{2x_i+x_{i+1}}{3}\right)+2\mathcal{C}\left(\frac{x_i+2x_{i+1}}{3}\right)+\mathcal{C}\left(x_{i+1}\right)\right)$$

and $R\left(\mathcal{C},{\rm Y}\right)$ is the error of approximation.

Proposition 1.

We let $\mathcal{C}$ be as in Theorem 1 and $n\in \mathbb{N}$. If ${\left|{\mathcal{C}}^{\prime \prime }\right|}^q$ is an s-convex function in the second sense for some fixed $s\in \left(0,1\right]$, we have

$$\left|R\left(\mathcal{C},{\rm Y}\right)\right|\le \underset{i=0}{\sum ^{n-1}}\frac{{\left(x_{i+1}-x_i\right)}^3}{54}\left(\frac{\left(2+2^{s+3}+3^{s+2}\right)s+\left(6+3\times 2^{s+3}-3^{s+3}\right)}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)\left(\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|+\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|\right).$$

Proof. 

Applying Theorem 1 with $\eta \left(e,l\right)=e-l$ and $m=1$ on $\left[x_i,x_{i+1}\right]$$\left(i=0,1,\cdots ,n-1\right)$ of partition ${\rm Y}$, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(x_i\right)+2\mathcal{C}\left(\frac{2x_i+x_{i+1}}{3}\right)+2\mathcal{C}\left(\frac{x_i+2x_{i+1}}{3}\right)+\mathcal{C}\left(x_{i+1}\right)\right)-\frac{1}{x_{i+1}-x_i}\stackrel{x_{i+1}}{\underset{x_i}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \underset{i=0}{\sum ^{n-1}}\frac{{\left(x_{i+1}-x_i\right)}^2}{54}\left(\frac{\left(2+2^{s+3}+3^{s+2}\right)s+\left(6+3\times 2^{s+3}-3^{s+3}\right)}{3^s\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)\left(\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|+\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|\right).\hfill \end{array}$$

We add the above inequalities for all $i=0,1,\cdots ,n-1$, and then multiply the resulting inequality by $\left(x_{i+1}-x_i\right)$. The desired result follows from the triangular inequality. □

Proposition 2.

We let $\mathcal{C}$ be as in Theorem 1 and $n\in \mathbb{N}$. If ${\left|{\mathcal{C}}^{\prime \prime }\right|}^q$ is convex function where $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, we have

$$\left|R\left(\mathcal{C},{\rm Y}\right)\right|\le \underset{i=0}{\sum ^{n-1}}\frac{{\left(x_{i+1}-x_i\right)}^3}{18}{\left(B\left(p+1,p+1\right)\right)}^{\frac{1}{p}}\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{2}\right).$$

Proof. 

Applying Corollary 3 on $\left[x_i,x_{i+1}\right]$$\left(i=0,1,\cdots ,n-1\right)$ of partition ${\rm Y}$, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(x_i\right)+2\mathcal{C}\left(\frac{2x_i+x_{i+1}}{3}\right)+2\mathcal{C}\left(\frac{x_i+2x_{i+1}}{3}\right)+\mathcal{C}\left(x_{i+1}\right)\right)-\frac{1}{x_{i+1}-x_i}\stackrel{x_{i+1}}{\underset{x_i}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \underset{i=0}{\sum ^{n-1}}\frac{{\left(x_{i+1}-x_i\right)}^2}{18}{\left(B\left(p+1,p+1\right)\right)}^{\frac{1}{p}}\left(\frac{{\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{2}\right).\hfill \end{array}$$

We add above inequalities for all $i=0,1,\cdots ,n-1$, and then multiply the resulting inequality by $\left(x_{i+1}-x_i\right)$. The desired result follows from the triangular inequality. □

Proposition 3.

We let $\mathcal{C}$ be as in Theorem 1 and $n\in \mathbb{N}$. If ${\left|{\mathcal{C}}^{\prime \prime }\right|}^q$ is convex function where $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, we have

$$\begin{array}{ccc}\hfill \left|R\left(\mathcal{C},{\rm Y}\right)\right|& \le & \underset{i=0}{\sum ^{n-1}}\frac{{\left(x_{i+1}-x_i\right)}^3}{162}{\left(\frac{3}{p+1}\right)}^{\frac{1}{p}}\left({\left(\frac{\left(3q+5\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +{\left(\frac{\left(2q+3\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+\left(q+3\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}\hfill \\ & & +\left.{\left(\frac{\left(q+1\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+\left(2q+5\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

Proof. 

Applying Corollary 6 on $\left[x_i,x_{i+1}\right]$$\left(i=0,1,\cdots ,n-1\right)$ of partition ${\rm Y}$, we obtain

$$\begin{array}{ccc}& & \left|\frac{1}{6}\left(\mathcal{C}\left(x_i\right)+2\mathcal{C}\left(\frac{2x_i+x_{i+1}}{3}\right)+2\mathcal{C}\left(\frac{x_i+2x_{i+1}}{3}\right)+\mathcal{C}\left(x_{i+1}\right)\right)-\frac{1}{x_{i+1}-x_i}\stackrel{x_{i+1}}{\underset{x_i}{\int }}\mathcal{C}\left(u\right)du\right|\hfill \\ & \le & \underset{i=0}{\sum ^{n-1}}\frac{{\left(x_{i+1}-x_i\right)}^2}{162}{\left(\frac{3}{p+1}\right)}^{\frac{1}{p}}\left({\left(\frac{\left(3q+5\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+{\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}\right.\hfill \\ & & +\left.{\left(\frac{\left(2q+3\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+\left(q+3\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}+{\left(\frac{\left(q+1\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_i\right)\right|}^q+\left(2q+5\right){\left|{\mathcal{C}}^{\prime \prime }\left(x_{i+1}\right)\right|}^q}{\left(q+1\right)\left(q+2\right)}\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

We add above inequalities for all $i=0,1,\cdots ,n-1$, and then multiply the resulting inequality by $\left(x_{i+1}-x_i\right)$. The desired result follows from the triangular inequality. □

For arbitrary real numbers $l,e,t$, we have:

The arithmetic mean: $\mathcal{A}\left(l,e\right)=\frac{l+e}{2}$ and $\mathcal{A}\left(l,e,t\right)=\frac{l+e+t}{3}$.

The p-logarithmic mean: ${\mathcal{L}}_p\left(l,e\right)={\left(\frac{e^{p+1}-l^{p+1}}{\left(p+1\right)\left(e-l\right)}\right)}^{\frac{1}{p}}$, $l,e>0,l\ne e$ and $p\in \mathbb{R}\setminus \left\{-1,0\right\}$.

Proposition 4.

We let $l,e\in \mathbb{R}$ with $0<l<e$, then we have

$$\left|\mathcal{A}\left(l^3,e^3\right)+{\mathcal{A}}^3\left(l,l,e\right)+{\mathcal{A}}^3\left(l,e,e\right)-3{\mathcal{L}}_3^3\left(l,e\right)\right|\le \frac{{\left(e-l\right)}^2}{6}{\left(\frac{l^q+e^q}{2}\right)}^{\frac{1}{q}}.$$

Proof. 

The assertion follows from Corollary 5, with $q\ge 2$, applied to function $f\left(x\right)=x^3$. □