• Research
• Open Access

# Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex

Journal of Inequalities and Applications20132013:451

https://doi.org/10.1186/1029-242X-2013-451

• Accepted: 26 September 2013
• Published:

## Abstract

In the paper, the authors establish some new inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.

MSC: 26D15, 26A51, 41A55.

## Keywords

• convex function
• third derivative

## 1 Introduction

It is common knowledge in mathematical analysis that a function $f:I\subseteq \mathbb{R}\to \mathbb{R}$ is said to be convex on an interval I if the inequality
$f\left(\lambda x+\left(1-\lambda \right)y\right)\le \lambda f\left(x\right)+\left(1-\lambda \right)f\left(y\right)$
(1.1)
is valid for all $x,y\in I$ and $\lambda \in \left[0,1\right]$. If $f:I\subseteq \mathbb{R}\to \mathbb{R}$ is a convex function on I and $a,b\in I$ with $a, then the double inequality
$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le \frac{f\left(a\right)+f\left(b\right)}{2}$
(1.2)

holds. This double inequality is known in the literature as Hermite-Hadamard’s integral inequality for convex functions. The definition of convex functions and Hermite-Hadamard’s integral inequality (1.2) have been generalized, refined, and extended by many mathematicians in a lot of references. Some of them may be recited as follows.

Theorem 1.1 ([, Theorems 2.2 and 2.3])

Let $f:{I}^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ be differentiable on ${I}^{\circ }$ and $a,b\in {I}^{\circ }$ with $a.
1. (1)
If $|{f}^{\prime }\left(x\right)|$ is a convex function on $\left[a,b\right]$, then
$\begin{array}{r}|\frac{f\left(a\right)+f\left(b\right)}{2}-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x|\\ \phantom{\rule{1em}{0ex}}\le \frac{b-a}{8}\left(|{f}^{\prime }\left(a\right)|+|{f}^{\prime }\left(b\right)|\right).\end{array}$
(1.3)

2. (2)
If ${|{f}^{\prime }\left(x\right)|}^{p/\left(p-1\right)}$ for $p>1$ is a convex function on $\left[a,b\right]$, then
$\begin{array}{r}|\frac{f\left(a\right)+f\left(b\right)}{2}-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x|\\ \phantom{\rule{1em}{0ex}}\le \frac{b-a}{2{\left(p+1\right)}^{1/p}}{\left[\frac{{|{f}^{\prime }\left(a\right)|}^{p/\left(p-1\right)}+{|{f}^{\prime }\left(b\right)|}^{p/\left(p-1\right)}}{2}\right]}^{\left(p-1\right)/p}.\end{array}$
(1.4)

Theorem 1.2 ([, Theorems 2.2 and 2.3])

Let $f:{I}^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on ${I}^{\circ }$ and $a,b\in {I}^{\circ }$ with $a. If ${|f|}^{p/\left(p-1\right)}$ for $p>1$ is convex on $\left[a,b\right]$, then
$\begin{array}{rcl}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x|& \le & \frac{b-a}{16}{\left(\frac{4}{p+1}\right)}^{1/p}\left[{\left({|{f}^{\prime }\left(a\right)|}^{p/\left(p-1\right)}+3{|{f}^{\prime }\left(b\right)|}^{p/\left(p-1\right)}\right)}^{\left(p-1\right)/p}\\ +{\left(3{|{f}^{\prime }\left(a\right)|}^{p/\left(p-1\right)}+{|{f}^{\prime }\left(b\right)|}^{p/\left(p-1\right)}\right)}^{\left(p-1\right)/p}\right]\end{array}$
(1.5)
and
$|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x|\le \frac{b-a}{4}{\left(\frac{4}{p+1}\right)}^{1/p}\left[|{f}^{\prime }\left(a\right)|+|{f}^{\prime }\left(b\right)|\right].$
(1.6)

Definition 1.1 ()

A function $f:I\subseteq \mathbb{R}\to \left[0,\mathrm{\infty }\right)$ is said to be quasi-convex if
$f\left(\lambda x+\left(1-\lambda \right)y\right)\le sup\left\{f\left(x\right),f\left(y\right)\right\}$
(1.7)

holds for all $x,y\in I$ and $\lambda \in \left[0,1\right]$.

Theorem 1.3 ([, Theorem 2])

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be differentiable on ${I}^{\circ }$ such that ${f}^{‴}\in L\left(\left[a,b\right]\right)$ and $a,b\in {I}^{\circ }$ with $a. If $|{f}^{‴}|$ is quasi-convex on $\left[a,b\right]$, then
$\begin{array}{r}|{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{b-a}{6}\left[f\left(a\right)+4f\left(\frac{a+b}{2}\right)+f\left(b\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{4}}{1152}\left[max\left\{|{f}^{‴}\left(a\right)|,|{f}^{‴}\left(\frac{a+b}{2}\right)|\right\}+max\left\{|{f}^{‴}\left(\frac{a+b}{2}\right)|,|{f}^{‴}\left(b\right)|\right\}\right].\end{array}$
(1.8)

Definition 1.2 ()

Let $s\in \left(0,1\right]$. A function $f:{\mathbb{R}}_{0}\to {\mathbb{R}}_{0}$ is said to be s-convex in the second sense if
$f\left(\lambda x+\left(1-\lambda \right)y\right)\le {\lambda }^{s}f\left(x\right)+{\left(1-\lambda \right)}^{s}f\left(y\right)$
(1.9)

for all $x,y\in I$ and $\lambda \in \left[0,1\right]$.

Theorem 1.4 ([, Theorem 3.1])

Let $f:I\subseteq {\mathbb{R}}_{0}\to \mathbb{R}$ be differentiable on ${I}^{\circ }$, $a,b\in {I}^{\circ }$ with $a, and ${f}^{‴}\in L\left(\left[a,b\right]\right)$. If $q\ge 1$ and $|{f}^{‴}|$ is s-convex in the second sense on $\left[a,b\right]$ for $s\in \left(0,1\right]$, then
$\begin{array}{r}|\frac{f\left(a\right)+f\left(b\right)}{2}-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{b-a}{12}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{192}{\left[\frac{{2}^{2-s}\left(s+6+{2}^{s+2}s\right)}{\left(s+2\right)\left(s+3\right)\left(s+4\right)}\right]}^{1/q}{\left[{|{f}^{‴}\left(a\right)|}^{q}+{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}.\end{array}$
(1.10)

In this paper, we will create some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.

## 2 Lemma

For establishing some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex, we need an integral identity below.

Lemma 2.1 Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be a three times differentiable mapping on ${I}^{\circ }$ and $a,b\in {I}^{\circ }$ with $a. If ${f}^{‴}\in L\left(\left[a,b\right]\right)$, then
$\begin{array}{r}f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]\\ \phantom{\rule{1em}{0ex}}=\frac{{\left(b-a\right)}^{3}}{96}\left[{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right){f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{2em}{0ex}}-{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right){f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right].\end{array}$
(2.1)
Proof Integrating by part and changing variable of definite integral yield
$\begin{array}{r}{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right){f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{1em}{0ex}}=-\frac{2}{b-a}{\int }_{0}^{1}\left(3{t}^{2}-6t+2\right){f}^{″}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{1em}{0ex}}=\frac{4}{{\left(b-a\right)}^{2}}\left[{f}^{\prime }\left(b\right)+2{f}^{\prime }\left(\frac{a+b}{2}\right)\right]+\frac{48}{{\left(b-a\right)}^{3}}{\int }_{0}^{1}\left(t-1\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}f\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)\\ \phantom{\rule{1em}{0ex}}=\frac{4}{{\left(b-a\right)}^{2}}\left[{f}^{\prime }\left(b\right)+2{f}^{\prime }\left(\frac{a+b}{2}\right)\right]+\frac{48}{{\left(b-a\right)}^{3}}f\left(\frac{a+b}{2}\right)\\ \phantom{\rule{2em}{0ex}}-\frac{48}{{\left(b-a\right)}^{3}}{\int }_{0}^{1}f\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\end{array}$
and
$\begin{array}{r}{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right){f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{1em}{0ex}}=\frac{2}{b-a}{\int }_{0}^{1}\left(3{t}^{2}-6t+2\right){f}^{″}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{1em}{0ex}}=\frac{4}{{\left(b-a\right)}^{2}}\left[{f}^{\prime }\left(a\right)+2{f}^{\prime }\left(\frac{a+b}{2}\right)\right]-\frac{48}{{\left(b-a\right)}^{3}}{\int }_{0}^{1}\left(t-1\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}f\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)\\ \phantom{\rule{1em}{0ex}}=\frac{4}{{\left(b-a\right)}^{2}}\left[{f}^{\prime }\left(a\right)+2{f}^{\prime }\left(\frac{a+b}{2}\right)\right]-\frac{48}{{\left(b-a\right)}^{3}}f\left(\frac{a+b}{2}\right)\\ \phantom{\rule{2em}{0ex}}+\frac{48}{{\left(b-a\right)}^{3}}{\int }_{0}^{1}f\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t.\end{array}$

Lemma 2.1 is thus proved. □

## 3 Hermite-Hadamard type inequalities for convex functions

Basing on Lemma 2.1, we now start out to establish some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.

Theorem 3.1 Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be three times differentiable on ${I}^{\circ }$ and ${f}^{‴}\in L\left(\left[a,b\right]\right)$ for $a,b\in {I}^{\circ }$ with $a. If ${|{f}^{‴}|}^{q}$ for $q\ge 1$ is convex on $\left[a,b\right]$, then
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{384}\left\{{\left[\frac{4{|{f}^{‴}\left(a\right)|}^{q}+11{|{f}^{‴}\left(b\right)|}^{q}}{15}\right]}^{1/q}+{\left[\frac{11{|{f}^{‴}\left(a\right)|}^{q}+4{|{f}^{‴}\left(b\right)|}^{q}}{15}\right]}^{1/q}\right\}.\end{array}$
(3.1)
Proof Since ${|{f}^{‴}|}^{q}$ is convex on $\left[a,b\right]$, by Lemma 2.1 and Hölder’s inequality, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}\left\{{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right)|{f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{2em}{0ex}}+{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right)|{f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right\}\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left[{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left[{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right){|{f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+{\left[{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right){|{f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}\right\}\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left(\frac{1}{4}\right)}^{1-1/q}\left\{\left[\frac{1}{2}{\int }_{0}^{1}t{\left(1-t\right)}^{2}\left(2-t\right){|{f}^{‴}\left(a\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ {\phantom{\rule{2em}{0ex}}+\frac{1}{2}{\int }_{0}^{1}t\left(1-{t}^{2}\right)\left(2-t\right){|{f}^{‴}\left(b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}+\left[\frac{1}{2}{\int }_{0}^{1}t\left(1-{t}^{2}\right)\left(2-t\right){|{f}^{‴}\left(a\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ {\phantom{\rule{2em}{0ex}}+\frac{1}{2}{\int }_{0}^{1}t{\left(1-t\right)}^{2}\left(2-t\right){|{f}^{‴}\left(b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}\right\}\\ \phantom{\rule{1em}{0ex}}=\frac{{\left(b-a\right)}^{3}}{384}\left\{{\left[\frac{4{|{f}^{‴}\left(a\right)|}^{q}+11{|{f}^{‴}\left(b\right)|}^{q}}{15}\right]}^{1/q}+{\left[\frac{11{|{f}^{‴}\left(a\right)|}^{q}+4{|{f}^{‴}\left(b\right)|}^{q}}{15}\right]}^{1/q}\right\}.\end{array}$

The proof of Theorem 3.1 is complete. □

Corollary 3.1 Under conditions of Theorem  3.1, if $q=1$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{384}\left[|{f}^{‴}\left(a\right)|+|{f}^{‴}\left(b\right)|\right].\end{array}$
(3.2)
Theorem 3.2 Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be three times differentiable on ${I}^{\circ }$ and ${f}^{‴}\in L\left(\left[a,b\right]\right)$ for $a,b\in {I}^{\circ }$ with $a. If ${|{f}^{‴}|}^{q}$ for $q>1$ is convex on $\left[a,b\right]$ and if $q\ge r$ and $s\ge 0$, then
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left[2B\left(\frac{2q-r-1}{q-1},\frac{2q-s-1}{q-1}\right)-B\left(\frac{3q-r-2}{q-1},\frac{2q-s-1}{q-1}\right)\right]}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{2}\right)}^{1/q}\left\{\left(\left[2B\left(r+1,s+2\right)-B\left(r+2,s+2\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(r+1,s+1\right)+B\left(r+2,s+1\right)-B\left(r+3,s+1\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left(\left[2B\left(r+1,s+1\right)+B\left(r+2,s+1\right)-B\left(r+3,s+1\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(r+1,s+2\right)-B\left(r+2,s+2\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right)}^{1/q}\right\},\end{array}$
where $B\left(x,y\right)$ is the classical Beta function, which may be defined for $\mathfrak{Re}\left(x\right)>0$ and $\mathfrak{Re}\left(y\right)>0$ by
$B\left(x,y\right)={\int }_{0}^{1}{t}^{x-1}{\left(1-t\right)}^{y-1}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t.$
(3.3)
Proof By Lemma 2.1, Hölder’s inequality, and the convexity of ${|{f}^{‴}|}^{q}$ on $\left[a,b\right]$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}\left\{{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right)|{f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{2em}{0ex}}+{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right)|{f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right\}\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left({\int }_{0}^{1}{t}^{\left(q-r\right)/\left(q-1\right)}{\left(1-t\right)}^{\left(q-s\right)/\left(q-1\right)}\left(2-t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right)}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left({\int }_{0}^{1}{t}^{r}{\left(1-t\right)}^{s}\left(2-t\right){|{f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}+{\left({\int }_{0}^{1}{t}^{r}{\left(1-t\right)}^{s}\left(2-t\right){|{f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right)}^{1/q}\right\}\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left[2B\left(\frac{2q-r-1}{q-1},\frac{2q-s-1}{q-1}\right)-B\left(\frac{3q-r-2}{q-1},\frac{2q-s-1}{q-1}\right)\right]}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{\left[\frac{1}{2}{\int }_{0}^{1}{t}^{r}{\left(1-t\right)}^{s+1}\left(2-t\right){|{f}^{‴}\left(a\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ {\phantom{\rule{2em}{0ex}}+\frac{1}{2}{\int }_{0}^{1}{t}^{r}\left(1+t\right){\left(1-t\right)}^{s}\left(2-t\right){|{f}^{‴}\left(b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\frac{1}{2}{\int }_{0}^{1}{t}^{r}\left(1+t\right){\left(1-t\right)}^{s}\left(2-t\right){|{f}^{‴}\left(a\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ {\phantom{\rule{2em}{0ex}}+\frac{1}{2}{\int }_{0}^{1}{t}^{r}{\left(1-t\right)}^{s+1}\left(2-t\right){|{f}^{‴}\left(b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}\right\}\\ \phantom{\rule{1em}{0ex}}=\frac{{\left(b-a\right)}^{3}}{96}{\left[2B\left(\frac{2q-r-1}{q-1},\frac{2q-s-1}{q-1}\right)-B\left(\frac{3q-r-2}{q-1},\frac{2q-s-1}{q-1}\right)\right]}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×{\left(\frac{1}{2}\right)}^{1/q}\left\{\left[\left[2B\left(r+1,s+2\right)-B\left(r+2,s+2\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(r+1,s+1\right)+B\left(r+2,s+1\right)-B\left(r+3,s+1\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\left[2B\left(r+1,s+1\right)+B\left(r+2,s+1\right)-B\left(r+3,s+1\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(r+1,s+2\right)-B\left(r+2,s+2\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\}.\end{array}$

The proof of Theorem 3.2 is completed. □

Corollary 3.2 Under conditions of Theorem  3.2,
1. (1)
if $r=0$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}\left[2B\left(\frac{2q-1}{q-1},\frac{2q-s-1}{q-1}\right)\\ {\phantom{\rule{2em}{0ex}}-B\left(\frac{3q-2}{q-1},\frac{2q-s-1}{q-1}\right)\right]}^{1-1/q}{\left(\frac{1}{2\left(s+1\right)\left(s+2\right)\left(s+3\right)}\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left[\left(s+1\right)\left(2s+5\right){|{f}^{‴}\left(a\right)|}^{q}+\left(2{s}^{2}+11s+13\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+{\left[\left(2{s}^{2}+11s+13\right){|{f}^{‴}\left(a\right)|}^{q}+\left(s+1\right)\left(2s+5\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\};\end{array}$

2. (2)
if $s=0$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}\left[2B\left(\frac{2q-r-1}{q-1},\frac{2q-1}{q-1}\right)\\ {\phantom{\rule{2em}{0ex}}-B\left(\frac{3q-r-2}{q-1},\frac{2q-1}{q-1}\right)\right]}^{1-1/q}{\left[\frac{1}{2\left(r+1\right)\left(r+2\right)\left(r+3\right)}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left[\left(r+5\right){|{f}^{‴}\left(a\right)|}^{q}+\left(2{r}^{2}+11r+13\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+{\left[\left(2{r}^{2}+11r+13\right){|{f}^{‴}\left(a\right)|}^{q}+\left(r+5\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\};\end{array}$

3. (3)
if $r=s=0$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left[2B\left(\frac{2q-1}{q-1},\frac{2q-1}{q-1}\right)-B\left(\frac{3q-2}{q-1},\frac{2q-1}{q-1}\right)\right]}^{1-1/q}{\left(\frac{3}{2}\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left[\frac{5{|{f}^{‴}\left(a\right)|}^{q}+13{|{f}^{‴}\left(b\right)|}^{q}}{18}\right]}^{1/q}+{\left[\frac{13{|{f}^{‴}\left(a\right)|}^{q}+5{|{f}^{‴}\left(b\right)|}^{q}}{18}\right]}^{1/q}\right\};\end{array}$

4. (4)
if $r=q$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left[\frac{\left(5q-2s-3\right)\left(q-1\right)}{{\left(q-s\right)}^{2}+\left(5q-3s-2\right)\left(q-1\right)}\right]}^{1-1/q}{\left(\frac{1}{2}\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{\left[\left[2B\left(q+1,s+2\right)-B\left(q+2,s+2\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(q+1,s+1\right)+B\left(q+2,s+1\right)-B\left(q+3,s+1\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\left[2B\left(q+1,s+1\right)+B\left(q+2,s+1\right)-B\left(q+3,s+1\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(q+1,s+2\right)-B\left(q+2,s+2\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\};\end{array}$

5. (5)
if $s=q$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left(\frac{\left(q-1\right)\left(4q-r-3\right)}{\left(2q-r-1\right)\left(3q-r-2\right)}\right)}^{1-1/q}{\left(\frac{1}{2}\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{\left[\left[2B\left(r+1,q+2\right)-B\left(r+2,q+2\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(r+1,q+1\right)+B\left(r+2,q+1\right)-B\left(r+3,q+1\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\left[2B\left(r+1,q+1\right)+B\left(r+2,q+1\right)-B\left(r+3,q+1\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(r+1,q+2\right)-B\left(r+2,q+2\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\};\end{array}$

6. (6)
if $r=s=q$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{3}^{-1/q}{\left(b-a\right)}^{3}}{64}\left\{\left[\left[2B\left(q+1,q+2\right)-B\left(q+2,q+2\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(q+1,q+1\right)+B\left(q+2,q+1\right)-B\left(q+3,q+1\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\left[2B\left(q+1,q+1\right)+B\left(q+2,q+1\right)-B\left(q+3,q+1\right)\right]{|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left[2B\left(q+1,q+2\right)-B\left(q+2,q+2\right)\right]{|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\}.\end{array}$

Theorem 3.3 Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be three times differentiable on ${I}^{\circ }$ and ${f}^{‴}\in L\left(\left[a,b\right]\right)$ for $a,b\in {I}^{\circ }$ with $a. If ${|{f}^{‴}|}^{q}$ is convex on $\left[a,b\right]$ for $q>1$ and $q\ge \ell \ge 0$, then
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left[\frac{\left({2}^{\xi +2}+1\right)\xi -{2}^{\xi +2}+5}{{\xi }^{3}+6{\xi }^{2}+11\xi +6}\right]}^{1-1/q}{\left[\frac{1}{\left(\ell +1\right)\left(\ell +2\right)\left(\ell +3\right)\left(\ell +4\right)}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{\left[\left({2}^{\ell +1}{\ell }^{2}-\left({2}^{\ell +1}+1\right)\ell +{2}^{\ell +3}-7\right){|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left(\left({2}^{\ell +1}+1\right){\ell }^{2}+2\left(7×{2}^{\ell }+5\right)\ell -3×{2}^{\ell +3}+27\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\left(\left({2}^{\ell +1}+1\right){\ell }^{2}+2\left(7×{2}^{\ell }+5\right)\ell -3×{2}^{\ell +3}+27\right){|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left({2}^{\ell +1}{\ell }^{2}-\left({2}^{\ell +1}+1\right)\ell +{2}^{\ell +3}-7\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\},\end{array}$

where $\xi =\frac{q-\ell }{q-1}$.

Proof By Lemma 2.1, Hölder’s inequality, and the convexity of ${|{f}^{‴}|}^{q}$ on $\left[a,b\right]$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}\left\{{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right)|{f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\\ \phantom{\rule{2em}{0ex}}+{\int }_{0}^{1}t\left(1-t\right)\left(2-t\right)|{f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right\}\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left({\int }_{0}^{1}t\left(1-t\right){\left(2-t\right)}^{\left(q-\ell \right)/\left(q-1\right)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right)}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left({\int }_{0}^{1}t\left(1-t\right){\left(2-t\right)}^{\ell }{|{f}^{‴}\left(\frac{1-t}{2}a+\frac{1+t}{2}b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right)}^{1/q}\\ \phantom{\rule{2em}{0ex}}+{\left({\int }_{0}^{1}t\left(1-t\right){\left(2-t\right)}^{\ell }{|{f}^{‴}\left(\frac{1+t}{2}a+\frac{1-t}{2}b\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right)}^{1/q}\right\}\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left({\int }_{0}^{1}t\left(1-t\right){\left(2-t\right)}^{\left(q-\ell \right)/\left(q-1\right)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right)}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left[\frac{1}{2}{\int }_{0}^{1}t\left(1-t\right){\left(2-t\right)}^{\ell }\left[\left(1-t\right){|{f}^{‴}\left(a\right)|}^{q}+\left(1+t\right){|{f}^{‴}\left(b\right)|}^{q}\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+{\left[\frac{1}{2}{\int }_{0}^{1}t\left(1-t\right){\left(2-t\right)}^{\ell }\left[\left(1+t\right){|{f}^{‴}\left(a\right)|}^{q}+\left(1-t\right){|{f}^{‴}\left(b\right)|}^{q}\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\right]}^{1/q}\right\}\\ \phantom{\rule{1em}{0ex}}=\frac{{\left(b-a\right)}^{3}}{96}{\left[\frac{\left({2}^{\xi +2}+1\right)\xi -{2}^{\xi +2}+5}{{\xi }^{3}+6{\xi }^{2}+11\xi +6}\right]}^{1-1/q}{\left[\frac{1}{\left(\ell +1\right)\left(\ell +2\right)\left(\ell +3\right)\left(\ell +4\right)}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{\left[\left({2}^{\ell +1}{\ell }^{2}-\left({2}^{\ell +1}+1\right)\ell +{2}^{\ell +3}-7\right){|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left(\left({2}^{\ell +1}+1\right){\ell }^{2}+2\left(7×{2}^{\ell }+5\right)\ell -3×{2}^{\ell +3}+27\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\left(\left({2}^{\ell +1}+1\right){\ell }^{2}+2\left(7×{2}^{\ell }+5\right)\ell -3×{2}^{\ell +3}+27\right){|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left({2}^{\ell +1}{\ell }^{2}-\left({2}^{\ell +1}+1\right)\ell +{2}^{\ell +3}-7\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\}.\end{array}$

The proof of Theorem 3.3 is complete. □

Corollary 3.3 Under conditions of Theorem  3.3.
1. (1)
if $\ell =0$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left(\frac{1}{6}\right)}^{1/q}{\left[\frac{{\left(q-1\right)}^{2}\left(6q+{2}^{\left(3q-2\right)/\left(q-1\right)}-5\right)}{\left(2q-1\right)\left(3q-2\right)\left(4q-3\right)}\right]}^{1-1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{{\left[\frac{{|{f}^{‴}\left(a\right)|}^{q}+3{|{f}^{‴}\left(b\right)|}^{q}}{4}\right]}^{1/q}+{\left[\frac{3{|{f}^{‴}\left(a\right)|}^{q}+{|{f}^{‴}\left(b\right)|}^{q}}{4}\right]}^{1/q}\right\},\end{array}$
(3.4)

2. (2)
if $\ell =q$, we have
$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+\frac{b-a}{24}\left[{f}^{\prime }\left(b\right)-{f}^{\prime }\left(a\right)\right]|\\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{3}}{96}{\left(\frac{1}{6}\right)}^{1-1/q}{\left[\frac{1}{\left(q+1\right)\left(q+2\right)\left(q+3\right)\left(q+4\right)}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}×\left\{\left[\left({2}^{q+1}{q}^{2}-\left({2}^{q+1}+1\right)q+{2}^{q+3}-7\right){|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left(\left({2}^{q+1}+1\right){q}^{2}+2\left(7×{2}^{q}+5\right)q-3×{2}^{q+3}+27\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\\ \phantom{\rule{2em}{0ex}}+\left[\left(\left({2}^{q+1}+1\right){q}^{2}+2\left(7×{2}^{q}+5\right)q-3×{2}^{q+3}+27\right){|{f}^{‴}\left(a\right)|}^{q}\\ {\phantom{\rule{2em}{0ex}}+\left({2}^{q+1}{q}^{2}-\left({2}^{q+1}+1\right)q+{2}^{q+3}-7\right){|{f}^{‴}\left(b\right)|}^{q}\right]}^{1/q}\right\}.\end{array}$

## Declarations

### Acknowledgements

The authors appreciate the editor and anonymous referees for their careful reading, helpful comments on, and valuable suggestions to the original version of this manuscript. This work was partially supported by the NNSF of China under Grant No. 11361038, by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, and by the Science Research Funding of the Inner Mongolia University for Nationalities under Grant No. NMD1225, China.

## Authors’ Affiliations

(1)
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China
(2)
Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300160, China

## References 