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# Some Hermite-Hadamard type inequalities for n-time differentiable $\left(\alpha ,m\right)$-convex functions

Journal of Inequalities and Applications20122012:267

https://doi.org/10.1186/1029-242X-2012-267

• Received: 12 June 2012
• Accepted: 7 November 2012
• Published:

## Abstract

In the paper, the famous Hermite-Hadamard integral inequality for convex functions is generalized to and refined as inequalities for n-time differentiable functions which are $\left(\alpha ,m\right)$-convex.

MSC: 26D15, 26A51, 41A55.

## Keywords

• Hermite-Hadamard’s integral inequality
• differentiable function
• $\left(\alpha ,m\right)$-convex function

## 1 Introduction

Throughout this paper, we adopt the following notations:
$\mathbb{R}=\left(-\mathrm{\infty },\mathrm{\infty }\right),\phantom{\rule{2em}{0ex}}{\mathbb{R}}_{0}=\left[0,\mathrm{\infty }\right),\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{\mathbb{R}}_{+}=\left(0,\mathrm{\infty }\right).$
(1.1)

We recall some definitions of several convex functions.

Definition 1.1 A function $f:I\subseteq \mathbb{R}\to \mathbb{R}$ is said to be convex if
$f\left(\lambda x+\left(1-\lambda \right)y\right)\le \lambda f\left(x\right)+\left(1-\lambda \right)f\left(y\right)$
(1.2)

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

Definition 1.2 ()

For $f:\left[0,b\right]\to \mathbb{R}$ and $m\in \left(0,1\right]$, if
$f\left(\lambda x+m\left(1-\lambda \right)y\right)\le \lambda f\left(x\right)+m\left(1-\lambda \right)f\left(y\right)$
(1.3)

is valid for all $x,y\in \left[0,b\right]$ and $\lambda \in \left[0,1\right]$, then we say that $f\left(x\right)$ is an m-convex function on $\left[0,b\right]$.

Definition 1.3 ()

For $f:\left[0,b\right]\to \mathbb{R}$ and $\alpha ,m\in \left(0,1\right]$, if
$f\left(\lambda x+m\left(1-\lambda \right)y\right)\le {\lambda }^{\alpha }f\left(x\right)+m\left(1-{\lambda }^{\alpha }\right)f\left(y\right)$
(1.4)

is valid for all $x,y\in \left[0,b\right]$ and $\lambda \in \left[0,1\right]$, then we say that $f\left(x\right)$ is an $\left(\alpha ,m\right)$-convex function on $\left[0,b\right]$.

In recent decades, plenty of inequalities of Hermite-Hadamard type for various kinds of convex functions have been established. Some of them may be reformulated as follows.

Theorem 1.1 ([, Theorem 2.2])

Let $f:{I}^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping and $a,b\in {I}^{\circ }$ with $a. If $|{f}^{\prime }\left(x\right)|$ is convex on $\left[a,b\right]$, then
$|\frac{f\left(a\right)+f\left(b\right)}{2}-\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x|\le \frac{\left(b-a\right)\left(|{f}^{\prime }\left(a\right)|+|{f}^{\prime }\left(b\right)|\right)}{8}.$
(1.5)

Theorem 1.2 ([, Theorem 2])

Let $f:{\mathbb{R}}_{0}\to \mathbb{R}$ be m-convex and $m\in \left(0,1\right]$. If $f\in L\left[a,b\right]$ for $0\le a, then
$\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\le min\left\{\frac{f\left(a\right)+mf\left(b/m\right)}{2},\frac{mf\left(a/m\right)+f\left(b\right)}{2}\right\}.$
(1.6)

Theorem 1.3 ([, Theorem 2.2])

Let $I\supseteq {\mathbb{R}}_{0}$ be an open interval and let $f:I\to \mathbb{R}$ be a differentiable function such that ${f}^{\prime }\in L\left[a,b\right]$ for $0\le a. If ${|{f}^{\prime }\left(x\right)|}^{q}$ is m-convex on $\left[a,b\right]$ for some $m\in \left(0,1\right]$ and $q\ge 1$, then

Theorem 1.4 ([, Theorem 3.1])

Let $I\supseteq {\mathbb{R}}_{0}$ be an open interval and let $f:I\to \mathbb{R}$ be a differentiable function such that ${f}^{\prime }\in L\left[a,b\right]$ for $0\le a. If ${\left[{f}^{\prime }\left(x\right)\right]}^{q}$ is $\left(\alpha ,m\right)$-convex on $\left[a,b\right]$ for some $\alpha ,m\in \left(0,1\right]$ and $q\ge 1$, then
where
${v}_{1}=\frac{1}{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\alpha +\frac{1}{{2}^{\alpha }}\right)$
(1.8)
and
${v}_{2}=\frac{1}{\left(\alpha +1\right)\left(\alpha +2\right)}\left(\frac{{\alpha }^{2}+\alpha +2}{2}-\frac{1}{{2}^{\alpha }}\right).$
(1.9)

For more and detailed information on this topic, please refer to the monograph  and newly published papers .

In this paper, we establish some Hermite-Hadamard type integral inequalities for n-time differentiable functions which are $\left(\alpha ,m\right)$-convex.

## 2 A lemma

In order to find inequalities of Hermite-Hadamard type for $\left(\alpha ,m\right)$-convex functions, we need the following lemma.

Lemma 2.1 ([, Lemma 2.1] or [, Lemma 2.1])

Let $f:\left[a,b\right]\subseteq \mathbb{R}\to \mathbb{R}$ be an n-time differentiable function such that ${f}^{\left(n-1\right)}\left(x\right)$ for $n\in \mathbb{N}$ is absolutely continuous on $\left[a,b\right]$. Then the identity
$\begin{array}{rcl}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x& =& \sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)\\ +{\left(-1\right)}^{n}{\int }_{a}^{b}{K}_{n}\left(t,x\right){f}^{\left(n\right)}\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\end{array}$
(2.1)
holds for all $t\in \left[a,b\right]$, where the kernel ${K}_{n}:\left[a,b\right]×\left[a,b\right]\to \mathbb{R}$ is defined by
${K}_{n}\left(t,x\right)=\left\{\begin{array}{cc}\frac{{\left(x-a\right)}^{n}}{n!},\hfill & x\in \left[a,t\right],\hfill \\ \frac{{\left(x-b\right)}^{n}}{n!},\hfill & x\in \left[t,b\right].\hfill \end{array}$
(2.2)

## 3 Hermite-Hadamard type inequalities for $\left(\alpha ,m\right)$-convex functions

We now set off to establish some new integral inequalities of Hermite-Hadamard type for n-time differentiable $\left(\alpha ,m\right)$-convex functions.

Theorem 3.1 Let $f:{\mathbb{R}}_{0}\to \mathbb{R}$ be an n-time differentiable function for $n\in \mathbb{N}$ and let $0\le a and $\alpha ,m\in \left(0,1\right]$. If ${f}^{\left(n\right)}\left(x\right)\in L\left[a,\frac{b}{m}\right]$ and ${|{f}^{\left(n\right)}\left(x\right)|}^{q}$ for $q\ge 1$ is $\left(\alpha ,m\right)$-convex on $\left[0,\frac{b}{m}\right]$, then
where $t\in \left[a,b\right]$ and $B\left(\alpha ,\beta \right)$ is the beta function
$B\left(\alpha ,\beta \right)={\int }_{0}^{1}{t}^{\alpha -1}{\left(1-t\right)}^{\beta -1}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t,\phantom{\rule{1em}{0ex}}\alpha ,\beta >0.$
(3.2)
Proof If $a, by Lemma 2.1, Hölder’s integral inequality, and the $\left(\alpha ,m\right)$-convexity of ${|{f}^{\left(n\right)}\left(x\right)|}^{q}$, we have
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}\left[{\int }_{a}^{t}{\left(x-a\right)}^{n}|{f}^{\left(n\right)}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}x+{\int }_{t}^{b}{\left(b-x\right)}^{n}|{f}^{\left(n\right)}\left(x\right)|\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right]\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}\left\{{\left[{\int }_{a}^{t}{\left(x-a\right)}^{n}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right]}^{1-1/q}{\left[{\int }_{a}^{t}{\left(x-a\right)}^{n}{|{f}^{\left(n\right)}\left(x\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left[{\int }_{t}^{b}{\left(b-x\right)}^{n}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right]}^{1-1/q}{\left[{\int }_{t}^{b}{\left(b-x\right)}^{n}{|{f}^{\left(n\right)}\left(x\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right]}^{1/q}\right\}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{1}{\left(b-a\right)n!}\left\{{\left[\frac{{\left(t-a\right)}^{n+1}}{n+1}\right]}^{1-1/q}\left[{\int }_{a}^{t}{\left(x-a\right)}^{n}|{f}^{\left(n\right)}\left(\frac{t-x}{t-a}a\hfill \\ \phantom{\rule{2em}{0ex}}+{{m\frac{x-a}{t-a}×\frac{t}{m}\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right]}^{1/q}+{\left[\frac{{\left(b-t\right)}^{n+1}}{n+1}\right]}^{1-1/q}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left[{\int }_{t}^{b}{\left(b-x\right)}^{n}{|{f}^{\left(n\right)}\left(\frac{b-x}{b-t}t+m\frac{x-t}{b-t}×\frac{b}{m}\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right]}^{1/q}\right\}\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}\left\{{\left[\frac{{\left(t-a\right)}^{n+1}}{n+1}\right]}^{1-1/q}\left({\int }_{a}^{t}{\left(x-a\right)}^{n}\left[{\left(\frac{t-x}{t-a}\right)}^{\alpha }{|{f}^{\left(n\right)}\left(a\right)|}^{q}\hfill \\ \phantom{\rule{2em}{0ex}}+{m\left(1-{\left(\frac{t-x}{t-a}\right)}^{\alpha }\right){|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right)}^{1/q}+{\left[\frac{{\left(b-t\right)}^{n+1}}{n+1}\right]}^{1-1/q}\hfill \\ \phantom{\rule{2em}{0ex}}×\left({\int }_{t}^{b}{\left(b-x\right)}^{n}\left[{\left(\frac{b-x}{b-t}\right)}^{\alpha }{|{f}^{\left(n\right)}\left(t\right)|}^{q}\hfill \\ \phantom{\rule{2em}{0ex}}+{m\left(1-{\left(\frac{b-x}{b-t}\right)}^{\alpha }\right){|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\right)}^{1/q}\right\}.\hfill \end{array}$
Substituting
$\begin{array}{c}{\int }_{a}^{t}{\left(x-a\right)}^{n}\left\{{\left(\frac{t-x}{t-a}\right)}^{\alpha }{|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left[1-{\left(\frac{t-x}{t-a}\right)}^{\alpha }\right]{|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right\}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(t-a\right)}^{n+1}}{n+1}\left[\alpha B\left(n+2,\alpha \right){|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left(1-\alpha B\left(n+2,\alpha \right)\right){|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]\hfill \end{array}$
and
$\begin{array}{c}{\int }_{t}^{b}{\left(b-x\right)}^{n}\left\{{\left(\frac{b-x}{b-t}\right)}^{\alpha }{|{f}^{\left(n\right)}\left(t\right)|}^{q}+m\left[1-{\left(\frac{b-x}{b-t}\right)}^{\alpha }\right]{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right\}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(b-t\right)}^{n+1}}{\left(n+1\right)\left(n+\alpha +1\right)}\left[\left(n+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+\alpha m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right]\hfill \end{array}$

into the above inequality leads to the inequality (3.1) for $t\in \left(a,b\right)$.

If $t=a$ or $t=b$, by virtue of Lemma 2.1 and the property that ${|{f}^{\left(n\right)}\left(x\right)|}^{q}$ is $\left(\alpha ,m\right)$-convex on $\left[0,\frac{b}{m}\right]$, we have

The inequality (3.1) for $t=a$ or $t=b$ follows. Theorem 3.1 is thus proved. □

Corollary 3.1 Under the conditions of Theorem  3.1,

(1) when $q=1$, we have
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)\left(n+1\right)!}\left\{{\left(t-a\right)}^{n+1}\left[\alpha B\left(n+2,\alpha \right)|{f}^{\left(n\right)}\left(a\right)|\hfill \\ \phantom{\rule{2em}{0ex}}+m\left(1-\alpha B\left(n+2,\alpha \right)\right)|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|\right]\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(b-t\right)}^{n+1}\left[\frac{1}{n+\alpha +1}\left(\left(n+1\right)|{f}^{\left(n\right)}\left(t\right)|+\alpha m|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|\right)\right]\right\};\hfill \end{array}$
(2) when $\alpha =1$, we have
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)\left(n+1\right)!}{\left(\frac{1}{n+2}\right)}^{1/q}\left\{{\left(t-a\right)}^{n+1}{\left[{|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left(n+1\right){|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(b-t\right)}^{n+1}{\left[\left(\left(n+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right)\right]}^{1/q}\right\};\hfill \end{array}$
(3) when $m=1$, we have
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)\left(n+1\right)!}\left\{{\left(t-a\right)}^{n+1}\left[\alpha B\left(n+2,\alpha \right){|{f}^{\left(n\right)}\left(a\right)|}^{q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(1-\alpha B\left(n+2,\alpha \right)\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(b-t\right)}^{n+1}{\left[\frac{1}{n+\alpha +1}\left(\left(n+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+\alpha {|{f}^{\left(n\right)}\left(b\right)|}^{q}\right)\right]}^{1/q}\right\};\hfill \end{array}$

Corollary 3.2 Under the conditions of Theorem  3.1,

Theorem 3.2 Let $t\in \left[a,b\right]$ and $f:{\mathbb{R}}_{0}\to \mathbb{R}$ be an n-time differentiable function for $n\in \mathbb{N}$, and let $0\le a and $\alpha ,m\in \left(0,1\right]$. If ${f}^{\left(n\right)}\left(x\right)\in L\left[a,\frac{b}{m}\right]$, ${|{f}^{\left(n\right)}\left(x\right)|}^{q}$ for $q>1$ is $\left(\alpha ,m\right)$-convex on $\left[0,\frac{b}{m}\right]$, and $nq\ge p\ge 0$, then
Proof When $a, by Lemma 2.1 and Hölder’s integral inequality, we have
where
${\int }_{a}^{t}{\left(x-a\right)}^{\left(nq-p\right)/\left(q-1\right)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\frac{q-1}{nq+q-p-1}{\left(t-a\right)}^{\left(nq+q-p-1\right)/\left(q-1\right)}$
(3.6)
and
${\int }_{t}^{b}{\left(b-x\right)}^{\left(nq-p\right)/\left(q-1\right)}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x=\frac{q-1}{nq+q-p-1}{\left(b-t\right)}^{\left(nq+q-p-1\right)/\left(q-1\right)}.$
(3.7)
Since ${|{f}^{\left(n\right)}\left(x\right)|}^{q}$ is $\left(\alpha ,m\right)$-convex on $\left[0,\frac{b}{m}\right]$, we have
$\begin{array}{c}{\int }_{a}^{t}{\left(x-a\right)}^{p}{|{f}^{\left(n\right)}\left(x\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{a}^{t}{\left(x-a\right)}^{p}\left\{{\left(\frac{t-x}{t-a}\right)}^{\alpha }{|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left[1-{\left(\frac{t-x}{t-a}\right)}^{\alpha }\right]{|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right\}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(t-a\right)}^{p+1}}{p+1}\left[\alpha B\left(p+2,\alpha \right){|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left(1-\alpha B\left(p+2,\alpha \right)\right){|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]\hfill \end{array}$
and
$\begin{array}{c}{\int }_{t}^{b}{\left(b-x\right)}^{p}{|{f}^{\left(n\right)}\left(x\right)|}^{q}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\hfill \\ \phantom{\rule{1em}{0ex}}\le {\int }_{t}^{b}{\left(b-x\right)}^{p}\left\{{\left(\frac{b-x}{b-t}\right)}^{\alpha }{|{f}^{\left(n\right)}\left(t\right)|}^{q}+m\left[1-{\left(\frac{b-x}{b-t}\right)}^{\alpha }\right]{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right\}\phantom{\rule{0.2em}{0ex}}\mathrm{d}x\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\left(b-t\right)}^{p+1}}{\left(p+1\right)\left(p+\alpha +1\right)}\left[\left(p+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+\alpha m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right].\hfill \end{array}$

Hence, the inequality (3.4) follows.

When $t=a$ or $t=b$, the proof of the inequality (3.4) is similar to the above argument. The proof of Theorem 3.2 is complete. □

Corollary 3.3 Under the conditions of Theorem  3.2,

(1) if $\alpha =1$, then
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}{\left(\frac{q-1}{nq+q-p-1}\right)}^{1-1/q}{\left[\frac{1}{\left(p+1\right)\left(p+2\right)}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}×\left\{{\left(t-a\right)}^{n+1}{\left[{|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left(p+1\right){|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(b-t\right)}^{n+1}{\left[\left(\left(p+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right)\right]}^{1/q}\right\};\hfill \end{array}$
(2) if $m=1$, then
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}{\left(\frac{q-1}{nq+q-p-1}\right)}^{1-1/q}{\left(\frac{1}{p+1}\right)}^{1/q}\left\{{\left(t-a\right)}^{n+1}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left[\alpha B\left(p+2,\alpha \right){|{f}^{\left(n\right)}\left(a\right)|}^{q}+\left(1-\alpha B\left(p+2,\alpha \right)\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(b-t\right)}^{n+1}{\left[\frac{1}{p+\alpha +1}\left(\left(p+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+\alpha {|{f}^{\left(n\right)}\left(b\right)|}^{q}\right)\right]}^{1/q}\right\};\hfill \end{array}$
(3) if $m=\alpha =1$, we have
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}{\left(\frac{q-1}{nq+q-p-1}\right)}^{1-1/q}{\left[\frac{1}{\left(p+1\right)\left(p+2\right)}\right]}^{1/q}\left\{{\left(t-a\right)}^{n+1}\left[{|{f}^{\left(n\right)}\left(a\right)|}^{q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(p+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}\right]}^{1/q}+{\left(b-t\right)}^{n+1}{\left[\left(p+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+{|{f}^{\left(n\right)}\left(b\right)|}^{q}\right]}^{1/q}\right\}.\hfill \end{array}$

Corollary 3.4 Under the conditions of Theorem  3.2,

(2) if $t=\frac{a+b}{2}$, then
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\sum _{k=0}^{n-1}\frac{\left[1+{\left(-1\right)}^{k}\right]{\left(b-a\right)}^{k}}{{2}^{k+1}\left(k+1\right)!}{f}^{\left(k\right)}\left(\frac{a+b}{2}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{{\left(b-a\right)}^{n}}{{2}^{n+1}n!}{\left(\frac{q-1}{nq+q-p-1}\right)}^{1-1/q}{\left(\frac{1}{p+1}\right)}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}×\left\{{\left[\alpha B\left(p+2,\alpha \right){|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left(1-\alpha B\left(p+2,\alpha \right)\right){|{f}^{\left(n\right)}\left(\frac{a+b}{2m}\right)|}^{q}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left[\frac{1}{p+\alpha +1}\left(\left(p+1\right){|{f}^{\left(n\right)}\left(\frac{a+b}{2}\right)|}^{q}+\alpha m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right)\right]}^{1/q}\right\};\hfill \end{array}$

Corollary 3.5 Under the conditions of Theorem  3.2,

(1) if $p=0$, then
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}{\left(\frac{q-1}{nq+q-1}\right)}^{1-1/q}{\left[\frac{1}{\left(\alpha +1\right)\left(\alpha +2\right)}\right]}^{1/q}\left\{{\left(t-a\right)}^{n+1}\left[{|{f}^{\left(n\right)}\left(a\right)|}^{q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\alpha m{|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]}^{1/q}+{\left(b-t\right)}^{n+1}{\left[{|{f}^{\left(n\right)}\left(t\right)|}^{q}+\alpha m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right]}^{1/q}\right\};\hfill \end{array}$
(2) if $p=q$, then
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}{\left(\frac{q-1}{nq-1}\right)}^{1-1/q}{\left(\frac{1}{q+1}\right)}^{1/q}\left\{{\left(t-a\right)}^{n+1}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left[\alpha B\left(q+2,\alpha \right){|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left(1-\alpha B\left(q+2,\alpha \right)\right){|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(b-t\right)}^{n+1}{\left[\frac{1}{q+\alpha +1}\left(\left(q+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+\alpha m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right)\right]}^{1/q}\right\};\hfill \end{array}$
(3) if $p=nq$, then
$\begin{array}{c}|\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{\left(b-t\right)}^{k+1}+{\left(-1\right)}^{k}{\left(t-a\right)}^{k+1}}{\left(k+1\right)!}{f}^{\left(k\right)}\left(t\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le \frac{1}{\left(b-a\right)n!}{\left(\frac{1}{nq+1}\right)}^{1/q}\left\{{\left(t-a\right)}^{n+1}\hfill \\ \phantom{\rule{2em}{0ex}}×{\left[\alpha B\left(nq+2,\alpha \right){|{f}^{\left(n\right)}\left(a\right)|}^{q}+m\left(1-\alpha B\left(nq+2,\alpha \right)\right){|{f}^{\left(n\right)}\left(\frac{t}{m}\right)|}^{q}\right]}^{1/q}\hfill \\ \phantom{\rule{2em}{0ex}}+{\left(b-t\right)}^{n+1}{\left[\frac{1}{nq+\alpha +1}\left(\left(nq+1\right){|{f}^{\left(n\right)}\left(t\right)|}^{q}+\alpha m{|{f}^{\left(n\right)}\left(\frac{b}{m}\right)|}^{q}\right)\right]}^{1/q}\right\}.\hfill \end{array}$

## Declarations

### Acknowledgements

This work was supported partially by Science Research Funding of Inner Mongolia University for Nationalities under Grant No. NMD1103 and the National Natural Science Foundation of China under Grant No. 10962004.

## Authors’ Affiliations

(1)
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China
(2)
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China
(3)
Department of Mathematics, School of Science, Tianjin Polytechnic University, Tianjin City, 300387, China

## References 