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

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.

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

$$f(\lambda x+(1-\lambda )y)\le \lambda f(x)+(1-\lambda )f(y)$$

(1.1)

is valid for all $x,y\in I$ and $\lambda \in [0,1]$. If $f:I\subseteq \mathbb{R}\to \mathbb{R}$ is a convex function on I and $a,b\in I$ with $a<b$, then the double inequality

$$f\left(\frac{a+b}{2}\right)\le \frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x\le \frac{f(a)+f(b)}{2}$$

(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 ([[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 $a<b$.

• (1) If $|f^{\prime }(x)|$ is a convex function on $[a,b]$, then

  $$\begin{array}{r}|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x|\\ \le \frac{b-a}{8}(|f^{\prime }(a)|+|f^{\prime }(b)|).\end{array}$$

  (1.3)

• (2) If ${|f^{\prime }(x)|}^{p/(p-1)}$ for $p>1$ is a convex function on $[a,b]$, then

  $$\begin{array}{r}|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x|\\ \le \frac{b-a}{2{(p+1)}^{1/p}}{\left[\frac{{|f^{\prime }(a)|}^{p/(p-1)}+{|f^{\prime }(b)|}^{p/(p-1)}}{2}\right]}^{(p-1)/p}.\end{array}$$

  (1.4)

Theorem 1.2 ([[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 $a<b$. If ${|f|}^{p/(p-1)}$ for $p>1$ is convex on $[a,b]$, then

$$\begin{array}{rcl}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x|& \le & \frac{b-a}{16}{\left(\frac{4}{p+1}\right)}^{1/p}[{({|f^{\prime }(a)|}^{p/(p-1)}+3{|f^{\prime }(b)|}^{p/(p-1)})}^{(p-1)/p}\\ +{(3{|f^{\prime }(a)|}^{p/(p-1)}+{|f^{\prime }(b)|}^{p/(p-1)})}^{(p-1)/p}]\end{array}$$

(1.5)

and

$$|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x|\le \frac{b-a}{4}{\left(\frac{4}{p+1}\right)}^{1/p}[|f^{\prime }(a)|+|f^{\prime }(b)|].$$

(1.6)

Definition 1.1 ([3])

A function $f:I\subseteq \mathbb{R}\to [0,\mathrm{\infty })$ is said to be quasi-convex if

$$f(\lambda x+(1-\lambda )y)\le sup\{f(x),f(y)\}$$

(1.7)

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

Theorem 1.3 ([[4], Theorem 2])

Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be differentiable on $I^{\circ }$ such that $f^{‴}\in L([a,b])$ and $a,b\in I^{\circ }$ with $a<b$. If $|f^{‴}|$ is quasi-convex on $[a,b]$, then

$$\begin{array}{r}|{\int }_a^{b}f(x)\mathrm{d}x-\frac{b-a}{6}[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)]|\\ \le \frac{{(b-a)}^4}{1152}[max\{|f^{‴}(a)|,\left|f^{‴}\left(\frac{a+b}{2}\right)\right|\}+max\{\left|f^{‴}\left(\frac{a+b}{2}\right)\right|,|f^{‴}(b)|\}].\end{array}$$

(1.8)

Definition 1.2 ([5])

Let $s\in (0,1]$. A function $f:{\mathbb{R}}_0\to {\mathbb{R}}_0$ is said to be s-convex in the second sense if

$$f(\lambda x+(1-\lambda )y)\le {\lambda }^{s}f(x)+{(1-\lambda )}^{s}f(y)$$

(1.9)

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

Theorem 1.4 ([[6], Theorem 3.1])

Let $f:I\subseteq {\mathbb{R}}_0\to \mathbb{R}$ be differentiable on $I^{\circ }$, $a,b\in I^{\circ }$ with $a<b$, and $f^{‴}\in L([a,b])$. If $q\ge 1$ and $|f^{‴}|$ is s-convex in the second sense on $[a,b]$ for $s\in (0,1]$, then

$$\begin{array}{r}|\frac{f(a)+f(b)}{2}-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{b-a}{12}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{192}{\left[\frac{2^{2-s}(s+6+2^{s+2}s)}{(s+2)(s+3)(s+4)}\right]}^{1/q}{[{|f^{‴}(a)|}^q+{|f^{‴}(b)|}^q]}^{1/q}.\end{array}$$

(1.10)

For more information on Hermite-Hadamard type inequalities, please refer to [7–19], for example, and to monographs [20, 21] and related references therein.

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 $a<b$. If $f^{‴}\in L([a,b])$, then

$$\begin{array}{r}f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]\\ =\frac{{(b-a)}^3}{96}[{\int }_0^{1}t(1-t)(2-t)f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\mathrm{d}t\\ -{\int }_0^{1}t(1-t)(2-t)f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\mathrm{d}t].\end{array}$$

(2.1)

Proof Integrating by part and changing variable of definite integral yield

$$\begin{array}{r}{\int }_0^{1}t(1-t)(2-t)f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\mathrm{d}t\\ =-\frac{2}{b-a}{\int }_0^1(3t^2-6t+2)f^{″}(\frac{1-t}{2}a+\frac{1+t}{2}b)\mathrm{d}t\\ =\frac{4}{{(b-a)}^2}[f^{\prime }(b)+2f^{\prime }\left(\frac{a+b}{2}\right)]+\frac{48}{{(b-a)}^3}{\int }_0^1(t-1)\mathrm{d}f(\frac{1-t}{2}a+\frac{1+t}{2}b)\\ =\frac{4}{{(b-a)}^2}[f^{\prime }(b)+2f^{\prime }\left(\frac{a+b}{2}\right)]+\frac{48}{{(b-a)}^3}f\left(\frac{a+b}{2}\right)\\ -\frac{48}{{(b-a)}^3}{\int }_0^{1}f(\frac{1-t}{2}a+\frac{1+t}{2}b)\mathrm{d}t\end{array}$$

and

$$\begin{array}{r}{\int }_0^{1}t(1-t)(2-t)f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\mathrm{d}t\\ =\frac{2}{b-a}{\int }_0^1(3t^2-6t+2)f^{″}(\frac{1+t}{2}a+\frac{1-t}{2}b)\mathrm{d}t\\ =\frac{4}{{(b-a)}^2}[f^{\prime }(a)+2f^{\prime }\left(\frac{a+b}{2}\right)]-\frac{48}{{(b-a)}^3}{\int }_0^1(t-1)\mathrm{d}f(\frac{1+t}{2}a+\frac{1-t}{2}b)\\ =\frac{4}{{(b-a)}^2}[f^{\prime }(a)+2f^{\prime }\left(\frac{a+b}{2}\right)]-\frac{48}{{(b-a)}^3}f\left(\frac{a+b}{2}\right)\\ +\frac{48}{{(b-a)}^3}{\int }_0^{1}f(\frac{1+t}{2}a+\frac{1-t}{2}b)\mathrm{d}t.\end{array}$$

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 $f^{‴}\in L([a,b])$ for $a,b\in I^{\circ }$ with $a<b$. If ${|f^{‴}|}^q$ for $q\ge 1$ is convex on $[a,b]$, then

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{384}\{{\left[\frac{4{|f^{‴}(a)|}^q+11{|f^{‴}(b)|}^q}{15}\right]}^{1/q}+{\left[\frac{11{|f^{‴}(a)|}^q+4{|f^{‴}(b)|}^q}{15}\right]}^{1/q}\}.\end{array}$$

(3.1)

Proof Since ${|f^{‴}|}^q$ is convex on $[a,b]$, by Lemma 2.1 and Hölder’s inequality, we have

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}\{{\int }_0^{1}t(1-t)(2-t)\left|f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\right|\mathrm{d}t\\ +{\int }_0^{1}t(1-t)(2-t)\left|f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\right|\mathrm{d}t\}\\ \le \frac{{(b-a)}^3}{96}{[{\int }_0^{1}t(1-t)(2-t)\mathrm{d}t]}^{1-1/q}\\ \times \{{[{\int }_0^{1}t(1-t)(2-t){\left|f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\right|}^q\mathrm{d}t]}^{1/q}\\ +{[{\int }_0^{1}t(1-t)(2-t){\left|f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\right|}^q\mathrm{d}t]}^{1/q}\}\\ \le \frac{{(b-a)}^3}{96}{\left(\frac{1}{4}\right)}^{1-1/q}\left\{\right[\frac{1}{2}{\int }_0^{1}t{(1-t)}^2(2-t){|f^{‴}(a)|}^q\mathrm{d}t\\ {+\frac{1}{2}{\int }_0^{1}t(1-t^2)(2-t){|f^{‴}(b)|}^q\mathrm{d}t]}^{1/q}+[\frac{1}{2}{\int }_0^{1}t(1-t^2)(2-t){|f^{‴}(a)|}^q\mathrm{d}t\\ {+\frac{1}{2}{\int }_0^{1}t{(1-t)}^2(2-t){|f^{‴}(b)|}^q\mathrm{d}t]}^{1/q}\}\\ =\frac{{(b-a)}^3}{384}\{{\left[\frac{4{|f^{‴}(a)|}^q+11{|f^{‴}(b)|}^q}{15}\right]}^{1/q}+{\left[\frac{11{|f^{‴}(a)|}^q+4{|f^{‴}(b)|}^q}{15}\right]}^{1/q}\}.\end{array}$$

The proof of Theorem 3.1 is complete. □

Corollary 3.1 Under conditions of Theorem  3.1, if $q=1$, we have

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{384}[|f^{‴}(a)|+|f^{‴}(b)|].\end{array}$$

(3.2)

Theorem 3.2 Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be three times differentiable on $I^{\circ }$ and $f^{‴}\in L([a,b])$ for $a,b\in I^{\circ }$ with $a<b$. If ${|f^{‴}|}^q$ for $q>1$ is convex on $[a,b]$ and if $q\ge r$ and $s\ge 0$, then

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}{[2B(\frac{2q-r-1}{q-1},\frac{2q-s-1}{q-1})-B(\frac{3q-r-2}{q-1},\frac{2q-s-1}{q-1})]}^{1-1/q}\\ \times {\left(\frac{1}{2}\right)}^{1/q}\left\{\right([2B(r+1,s+2)-B(r+2,s+2)]{|f^{‴}(a)|}^q\\ {+[2B(r+1,s+1)+B(r+2,s+1)-B(r+3,s+1)]{|f^{‴}(b)|}^q)}^{1/q}\\ +([2B(r+1,s+1)+B(r+2,s+1)-B(r+3,s+1)]{|f^{‴}(a)|}^q\\ {+[2B(r+1,s+2)-B(r+2,s+2)]{|f^{‴}(b)|}^q)}^{1/q}\},\end{array}$$

where $B(x,y)$ is the classical Beta function, which may be defined for $\mathfrak{Re}(x)>0$ and $\mathfrak{Re}(y)>0$ by

$$B(x,y)={\int }_0^1{t}^{x-1}{(1-t)}^{y-1}\mathrm{d}t.$$

(3.3)

Proof By Lemma 2.1, Hölder’s inequality, and the convexity of ${|f^{‴}|}^q$ on $[a,b]$, we have

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}\{{\int }_0^{1}t(1-t)(2-t)\left|f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\right|\mathrm{d}t\\ +{\int }_0^{1}t(1-t)(2-t)\left|f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\right|\mathrm{d}t\}\\ \le \frac{{(b-a)}^3}{96}{({\int }_0^1{t}^{(q-r)/(q-1)}{(1-t)}^{(q-s)/(q-1)}(2-t)\mathrm{d}t)}^{1-1/q}\\ \times \{{({\int }_0^1{t}^r{(1-t)}^s(2-t){\left|f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\right|}^q\mathrm{d}t)}^{1/q}\\ +{({\int }_0^1{t}^r{(1-t)}^s(2-t){\left|f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\right|}^q\mathrm{d}t)}^{1/q}\}\\ \le \frac{{(b-a)}^3}{96}{[2B(\frac{2q-r-1}{q-1},\frac{2q-s-1}{q-1})-B(\frac{3q-r-2}{q-1},\frac{2q-s-1}{q-1})]}^{1-1/q}\\ \times \left\{\right[\frac{1}{2}{\int }_0^1{t}^r{(1-t)}^{s+1}(2-t){|f^{‴}(a)|}^q\mathrm{d}t\\ {+\frac{1}{2}{\int }_0^1{t}^r(1+t){(1-t)}^s(2-t){|f^{‴}(b)|}^q\mathrm{d}t]}^{1/q}\\ +[\frac{1}{2}{\int }_0^1{t}^r(1+t){(1-t)}^s(2-t){|f^{‴}(a)|}^q\mathrm{d}t\\ {+\frac{1}{2}{\int }_0^1{t}^r{(1-t)}^{s+1}(2-t){|f^{‴}(b)|}^q\mathrm{d}t]}^{1/q}\}\\ =\frac{{(b-a)}^3}{96}{[2B(\frac{2q-r-1}{q-1},\frac{2q-s-1}{q-1})-B(\frac{3q-r-2}{q-1},\frac{2q-s-1}{q-1})]}^{1-1/q}\\ \times {\left(\frac{1}{2}\right)}^{1/q}\left\{\right[[2B(r+1,s+2)-B(r+2,s+2)]{|f^{‴}(a)|}^q\\ {+[2B(r+1,s+1)+B(r+2,s+1)-B(r+3,s+1)]{|f^{‴}(b)|}^q]}^{1/q}\\ +[[2B(r+1,s+1)+B(r+2,s+1)-B(r+3,s+1)]{|f^{‴}(a)|}^q\\ {+[2B(r+1,s+2)-B(r+2,s+2)]{|f^{‴}(b)|}^q]}^{1/q}\}.\end{array}$$

The proof of Theorem 3.2 is completed. □

Corollary 3.2 Under conditions of Theorem  3.2,

• (1) if $r=0$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}[2B(\frac{2q-1}{q-1},\frac{2q-s-1}{q-1})\\ {-B(\frac{3q-2}{q-1},\frac{2q-s-1}{q-1})]}^{1-1/q}{\left(\frac{1}{2(s+1)(s+2)(s+3)}\right)}^{1/q}\\ \times \{{[(s+1)(2s+5){|f^{‴}(a)|}^q+(2s^2+11s+13){|f^{‴}(b)|}^q]}^{1/q}\\ +{[(2s^2+11s+13){|f^{‴}(a)|}^q+(s+1)(2s+5){|f^{‴}(b)|}^q]}^{1/q}\};\end{array}$$

• (2) if $s=0$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}[2B(\frac{2q-r-1}{q-1},\frac{2q-1}{q-1})\\ {-B(\frac{3q-r-2}{q-1},\frac{2q-1}{q-1})]}^{1-1/q}{\left[\frac{1}{2(r+1)(r+2)(r+3)}\right]}^{1/q}\\ \times \{{[(r+5){|f^{‴}(a)|}^q+(2r^2+11r+13){|f^{‴}(b)|}^q]}^{1/q}\\ +{[(2r^2+11r+13){|f^{‴}(a)|}^q+(r+5){|f^{‴}(b)|}^q]}^{1/q}\};\end{array}$$

• (3) if $r=s=0$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}{[2B(\frac{2q-1}{q-1},\frac{2q-1}{q-1})-B(\frac{3q-2}{q-1},\frac{2q-1}{q-1})]}^{1-1/q}{\left(\frac{3}{2}\right)}^{1/q}\\ \times \{{\left[\frac{5{|f^{‴}(a)|}^q+13{|f^{‴}(b)|}^q}{18}\right]}^{1/q}+{\left[\frac{13{|f^{‴}(a)|}^q+5{|f^{‴}(b)|}^q}{18}\right]}^{1/q}\};\end{array}$$

• (4) if $r=q$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}{\left[\frac{(5q-2s-3)(q-1)}{{(q-s)}^2+(5q-3s-2)(q-1)}\right]}^{1-1/q}{\left(\frac{1}{2}\right)}^{1/q}\\ \times \left\{\right[[2B(q+1,s+2)-B(q+2,s+2)]{|f^{‴}(a)|}^q\\ {+[2B(q+1,s+1)+B(q+2,s+1)-B(q+3,s+1)]{|f^{‴}(b)|}^q]}^{1/q}\\ +[[2B(q+1,s+1)+B(q+2,s+1)-B(q+3,s+1)]{|f^{‴}(a)|}^q\\ {+[2B(q+1,s+2)-B(q+2,s+2)]{|f^{‴}(b)|}^q]}^{1/q}\};\end{array}$$

• (5) if $s=q$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}{\left(\frac{(q-1)(4q-r-3)}{(2q-r-1)(3q-r-2)}\right)}^{1-1/q}{\left(\frac{1}{2}\right)}^{1/q}\\ \times \left\{\right[[2B(r+1,q+2)-B(r+2,q+2)]{|f^{‴}(a)|}^q\\ {+[2B(r+1,q+1)+B(r+2,q+1)-B(r+3,q+1)]{|f^{‴}(b)|}^q]}^{1/q}\\ +[[2B(r+1,q+1)+B(r+2,q+1)-B(r+3,q+1)]{|f^{‴}(a)|}^q\\ {+[2B(r+1,q+2)-B(r+2,q+2)]{|f^{‴}(b)|}^q]}^{1/q}\};\end{array}$$

• (6) if $r=s=q$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{3^{-1/q}{(b-a)}^3}{64}\left\{\right[[2B(q+1,q+2)-B(q+2,q+2)]{|f^{‴}(a)|}^q\\ {+[2B(q+1,q+1)+B(q+2,q+1)-B(q+3,q+1)]{|f^{‴}(b)|}^q]}^{1/q}\\ +[[2B(q+1,q+1)+B(q+2,q+1)-B(q+3,q+1)]{|f^{‴}(a)|}^q\\ {+[2B(q+1,q+2)-B(q+2,q+2)]{|f^{‴}(b)|}^q]}^{1/q}\}.\end{array}$$

Theorem 3.3 Let $f:I\subseteq \mathbb{R}\to \mathbb{R}$ be three times differentiable on $I^{\circ }$ and $f^{‴}\in L([a,b])$ for $a,b\in I^{\circ }$ with $a<b$. If ${|f^{‴}|}^q$ is convex on $[a,b]$ for $q>1$ and $q\ge \ell \ge 0$, then

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}{\left[\frac{(2^{\xi +2}+1)\xi -2^{\xi +2}+5}{{\xi }^3+6{\xi }^2+11\xi +6}\right]}^{1-1/q}{\left[\frac{1}{(\ell +1)(\ell +2)(\ell +3)(\ell +4)}\right]}^{1/q}\\ \times \left\{\right[(2^{\ell +1}{\ell }^2-(2^{\ell +1}+1)\ell +2^{\ell +3}-7){|f^{‴}(a)|}^q\\ {+((2^{\ell +1}+1){\ell }^2+2(7\times 2^{\ell }+5)\ell -3\times 2^{\ell +3}+27){|f^{‴}(b)|}^q]}^{1/q}\\ +[((2^{\ell +1}+1){\ell }^2+2(7\times 2^{\ell }+5)\ell -3\times 2^{\ell +3}+27){|f^{‴}(a)|}^q\\ {+(2^{\ell +1}{\ell }^2-(2^{\ell +1}+1)\ell +2^{\ell +3}-7){|f^{‴}(b)|}^q]}^{1/q}\},\end{array}$$

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

Proof By Lemma 2.1, Hölder’s inequality, and the convexity of ${|f^{‴}|}^q$ on $[a,b]$, we have

$$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}\{{\int }_0^{1}t(1-t)(2-t)\left|f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\right|\mathrm{d}t\\ +{\int }_0^{1}t(1-t)(2-t)\left|f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\right|\mathrm{d}t\}\\ \le \frac{{(b-a)}^3}{96}{({\int }_0^{1}t(1-t){(2-t)}^{(q-\ell )/(q-1)}\mathrm{d}t)}^{1-1/q}\\ \times \{{({\int }_0^{1}t(1-t){(2-t)}^{\ell }{\left|f^{‴}(\frac{1-t}{2}a+\frac{1+t}{2}b)\right|}^q\mathrm{d}t)}^{1/q}\\ +{({\int }_0^{1}t(1-t){(2-t)}^{\ell }{\left|f^{‴}(\frac{1+t}{2}a+\frac{1-t}{2}b)\right|}^q\mathrm{d}t)}^{1/q}\}\\ \le \frac{{(b-a)}^3}{96}{({\int }_0^{1}t(1-t){(2-t)}^{(q-\ell )/(q-1)}\mathrm{d}t)}^{1-1/q}\\ \times \{{[\frac{1}{2}{\int }_0^{1}t(1-t){(2-t)}^{\ell }[(1-t){|f^{‴}(a)|}^q+(1+t){|f^{‴}(b)|}^q]\mathrm{d}t]}^{1/q}\\ +{[\frac{1}{2}{\int }_0^{1}t(1-t){(2-t)}^{\ell }[(1+t){|f^{‴}(a)|}^q+(1-t){|f^{‴}(b)|}^q]\mathrm{d}t]}^{1/q}\}\\ =\frac{{(b-a)}^3}{96}{\left[\frac{(2^{\xi +2}+1)\xi -2^{\xi +2}+5}{{\xi }^3+6{\xi }^2+11\xi +6}\right]}^{1-1/q}{\left[\frac{1}{(\ell +1)(\ell +2)(\ell +3)(\ell +4)}\right]}^{1/q}\\ \times \left\{\right[(2^{\ell +1}{\ell }^2-(2^{\ell +1}+1)\ell +2^{\ell +3}-7){|f^{‴}(a)|}^q\\ {+((2^{\ell +1}+1){\ell }^2+2(7\times 2^{\ell }+5)\ell -3\times 2^{\ell +3}+27){|f^{‴}(b)|}^q]}^{1/q}\\ +[((2^{\ell +1}+1){\ell }^2+2(7\times 2^{\ell }+5)\ell -3\times 2^{\ell +3}+27){|f^{‴}(a)|}^q\\ {+(2^{\ell +1}{\ell }^2-(2^{\ell +1}+1)\ell +2^{\ell +3}-7){|f^{‴}(b)|}^q]}^{1/q}\}.\end{array}$$

The proof of Theorem 3.3 is complete. □

Corollary 3.3 Under conditions of Theorem  3.3.

• (1) if $\ell =0$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}{\left(\frac{1}{6}\right)}^{1/q}{\left[\frac{{(q-1)}^2(6q+2^{(3q-2)/(q-1)}-5)}{(2q-1)(3q-2)(4q-3)}\right]}^{1-1/q}\\ \times \{{\left[\frac{{|f^{‴}(a)|}^q+3{|f^{‴}(b)|}^q}{4}\right]}^{1/q}+{\left[\frac{3{|f^{‴}(a)|}^q+{|f^{‴}(b)|}^q}{4}\right]}^{1/q}\},\end{array}$$

  (3.4)

• (2) if $\ell =q$, we have

  $$\begin{array}{r}|f\left(\frac{a+b}{2}\right)-\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x+\frac{b-a}{24}[f^{\prime }(b)-f^{\prime }(a)]|\\ \le \frac{{(b-a)}^3}{96}{\left(\frac{1}{6}\right)}^{1-1/q}{\left[\frac{1}{(q+1)(q+2)(q+3)(q+4)}\right]}^{1/q}\\ \times \left\{\right[(2^{q+1}{q}^2-(2^{q+1}+1)q+2^{q+3}-7){|f^{‴}(a)|}^q\\ {+((2^{q+1}+1)q^2+2(7\times 2^q+5)q-3\times 2^{q+3}+27){|f^{‴}(b)|}^q]}^{1/q}\\ +[((2^{q+1}+1)q^2+2(7\times 2^q+5)q-3\times 2^{q+3}+27){|f^{‴}(a)|}^q\\ {+(2^{q+1}{q}^2-(2^{q+1}+1)q+2^{q+3}-7){|f^{‴}(b)|}^q]}^{1/q}\}.\end{array}$$