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

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 $(\alpha ,m)$-convex.

MSC: 26D15, 26A51, 41A55.

1 Introduction

Throughout this paper, we adopt the following notations:

$$\mathbb{R}=(-\mathrm{\infty },\mathrm{\infty }),{\mathbb{R}}_0=[0,\mathrm{\infty }),\text{and}{\mathbb{R}}_{+}=(0,\mathrm{\infty }).$$

(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

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

(1.2)

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

Definition 1.2 ([1])

For $f:[0,b]\to \mathbb{R}$ and $m\in (0,1]$, if

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

(1.3)

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

Definition 1.3 ([2])

For $f:[0,b]\to \mathbb{R}$ and $\alpha ,m\in (0,1]$, if

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

(1.4)

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

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 ([[3], Theorem 2.2])

Let $f:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping and $a,b\in I^{\circ }$ with $a<b$. If $|f^{\prime }(x)|$ is convex on $[a,b]$, then

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

(1.5)

Theorem 1.2 ([[4], Theorem 2])

Let $f:{\mathbb{R}}_0\to \mathbb{R}$ be m-convex and $m\in (0,1]$. If $f\in L[a,b]$ for $0\le a<b<\mathrm{\infty }$, then

$$\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x\le min\{\frac{f(a)+mf(b/m)}{2},\frac{mf(a/m)+f(b)}{2}\}.$$

(1.6)

Theorem 1.3 ([[2], 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 $f^{\prime }\in L[a,b]$ for $0\le a<b<\mathrm{\infty }$. If ${|f^{\prime }(x)|}^q$ is m-convex on $[a,b]$ for some $m\in (0,1]$ and $q\ge 1$, then

(1.7)

Theorem 1.4 ([[2], 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 $f^{\prime }\in L[a,b]$ for $0\le a<b<\mathrm{\infty }$. If ${[f^{\prime }(x)]}^q$ is $(\alpha ,m)$-convex on $[a,b]$ for some $\alpha ,m\in (0,1]$ and $q\ge 1$, then

where

$$v_1=\frac{1}{(\alpha +1)(\alpha +2)}(\alpha +\frac{1}{2^{\alpha }})$$

(1.8)

and

$$v_2=\frac{1}{(\alpha +1)(\alpha +2)}(\frac{{\alpha }^2+\alpha +2}{2}-\frac{1}{2^{\alpha }}).$$

(1.9)

For more and detailed information on this topic, please refer to the monograph [5] and newly published papers [6–16].

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

2 A lemma

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

Lemma 2.1 ([[17], Lemma 2.1] or [[18], Lemma 2.1])

Let $f:[a,b]\subseteq \mathbb{R}\to \mathbb{R}$ be an n-time differentiable function such that $f^{(n-1)}(x)$ for $n\in \mathbb{N}$ is absolutely continuous on $[a,b]$. Then the identity

$$\begin{array}{rcl}{\int }_a^{b}f(x)\mathrm{d}x& =& \sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)\\ +{(-1)}^n{\int }_a^b{K}_n(t,x)f^{(n)}(x)\mathrm{d}x\end{array}$$

(2.1)

holds for all $t\in [a,b]$, where the kernel $K_n:[a,b]\times [a,b]\to \mathbb{R}$ is defined by

$$K_n(t,x)=\{\begin{array}{cc}\frac{{(x-a)}^n}{n!},\hfill & x\in [a,t],\hfill \\ \frac{{(x-b)}^n}{n!},\hfill & x\in [t,b].\hfill \end{array}$$

(2.2)

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

We now set off to establish some new integral inequalities of Hermite-Hadamard type for n-time differentiable $(\alpha ,m)$-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 $0\le a<b<\mathrm{\infty }$ and $\alpha ,m\in (0,1]$. If $f^{(n)}(x)\in L[a,\frac{b}{m}]$ and ${|f^{(n)}(x)|}^q$ for $q\ge 1$ is $(\alpha ,m)$-convex on $[0,\frac{b}{m}]$, then

(3.1)

where $t\in [a,b]$ and $B(\alpha ,\beta )$ is the beta function

$$B(\alpha ,\beta )={\int }_0^1{t}^{\alpha -1}{(1-t)}^{\beta -1}\mathrm{d}t,\alpha ,\beta >0.$$

(3.2)

Proof If $a<t<b$, by Lemma 2.1, Hölder’s integral inequality, and the $(\alpha ,m)$-convexity of ${|f^{(n)}(x)|}^q$, we have

$$\begin{array}{c}|\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)|\hfill \\ \le \frac{1}{(b-a)n!}[{\int }_a^t{(x-a)}^n|f^{(n)}(x)|\mathrm{d}x+{\int }_t^b{(b-x)}^n|f^{(n)}(x)|\mathrm{d}x]\hfill \\ \le \frac{1}{(b-a)n!}\{{\left[{\int }_a^t{(x-a)}^n\mathrm{d}x\right]}^{1-1/q}{\left[{\int }_a^t{(x-a)}^n{|f^{(n)}(x)|}^q\mathrm{d}x\right]}^{1/q}\hfill \\ +{\left[{\int }_t^b{(b-x)}^n\mathrm{d}x\right]}^{1-1/q}{\left[{\int }_t^b{(b-x)}^n{|f^{(n)}(x)|}^q\mathrm{d}x\right]}^{1/q}\}\hfill \\ =\frac{1}{(b-a)n!}\left\{{\left[\frac{{(t-a)}^{n+1}}{n+1}\right]}^{1-1/q}\right[{\int }_a^t{(x-a)}^n\left|f^{(n)}\right(\frac{t-x}{t-a}a\hfill \\ +{{m\frac{x-a}{t-a}\times \frac{t}{m})|}^q\mathrm{d}x]}^{1/q}+{\left[\frac{{(b-t)}^{n+1}}{n+1}\right]}^{1-1/q}\hfill \\ \times {\left[{\int }_t^b{(b-x)}^n{\left|f^{(n)}(\frac{b-x}{b-t}t+m\frac{x-t}{b-t}\times \frac{b}{m})\right|}^q\mathrm{d}x\right]}^{1/q}\}\hfill \\ \le \frac{1}{(b-a)n!}\left\{{\left[\frac{{(t-a)}^{n+1}}{n+1}\right]}^{1-1/q}\right({\int }_a^t{(x-a)}^n[{\left(\frac{t-x}{t-a}\right)}^{\alpha }{|f^{(n)}(a)|}^q\hfill \\ +{m(1-{\left(\frac{t-x}{t-a}\right)}^{\alpha }){\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q]\mathrm{d}x)}^{1/q}+{\left[\frac{{(b-t)}^{n+1}}{n+1}\right]}^{1-1/q}\hfill \\ \times \left({\int }_t^b{(b-x)}^n\right[{\left(\frac{b-x}{b-t}\right)}^{\alpha }{|f^{(n)}(t)|}^q\hfill \\ +{m(1-{\left(\frac{b-x}{b-t}\right)}^{\alpha }){\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q]\mathrm{d}x)}^{1/q}\}.\hfill \end{array}$$

Substituting

$$\begin{array}{c}{\int }_a^t{(x-a)}^n\{{\left(\frac{t-x}{t-a}\right)}^{\alpha }{|f^{(n)}(a)|}^q+m[1-{\left(\frac{t-x}{t-a}\right)}^{\alpha }]{\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q\}\mathrm{d}x\hfill \\ =\frac{{(t-a)}^{n+1}}{n+1}[\alpha B(n+2,\alpha ){|f^{(n)}(a)|}^q+m(1-\alpha B(n+2,\alpha )){\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q]\hfill \end{array}$$

and

$$\begin{array}{c}{\int }_t^b{(b-x)}^n\{{\left(\frac{b-x}{b-t}\right)}^{\alpha }{|f^{(n)}(t)|}^q+m[1-{\left(\frac{b-x}{b-t}\right)}^{\alpha }]{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q\}\mathrm{d}x\hfill \\ =\frac{{(b-t)}^{n+1}}{(n+1)(n+\alpha +1)}[(n+1){|f^{(n)}(t)|}^q+\alpha m{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q]\hfill \end{array}$$

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

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

and

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

$$\begin{array}{c}|\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)|\hfill \\ \le \frac{1}{(b-a)(n+1)!}\left\{{(t-a)}^{n+1}\right[\alpha B(n+2,\alpha )|f^{(n)}(a)|\hfill \\ +m(1-\alpha B(n+2,\alpha ))\left|f^{(n)}\left(\frac{t}{m}\right)\right|]\hfill \\ +{(b-t)}^{n+1}\left[\frac{1}{n+\alpha +1}((n+1)|f^{(n)}(t)|+\alpha m\left|f^{(n)}\left(\frac{b}{m}\right)\right|)\right]\};\hfill \end{array}$$

(2) when $\alpha =1$, we have

$$\begin{array}{c}|\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)|\hfill \\ \le \frac{1}{(b-a)(n+1)!}{\left(\frac{1}{n+2}\right)}^{1/q}\{{(t-a)}^{n+1}{[{|f^{(n)}(a)|}^q+m(n+1){\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q]}^{1/q}\hfill \\ +{(b-t)}^{n+1}{\left[((n+1){|f^{(n)}(t)|}^q+m{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q)\right]}^{1/q}\};\hfill \end{array}$$

(3) when $m=1$, we have

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

(4) when $m=\alpha =q=1$, we have

Corollary 3.2 Under the conditions of Theorem  3.1,

(1) when $t=a$, we have

(3.3)

(2) when $t=\frac{a+b}{2}$, we have

(3) when $t=b$, we have

Theorem 3.2 Let $t\in [a,b]$ and $f:{\mathbb{R}}_0\to \mathbb{R}$ be an n-time differentiable function for $n\in \mathbb{N}$, and let $0\le a<b<\mathrm{\infty }$ and $\alpha ,m\in (0,1]$. If $f^{(n)}(x)\in L[a,\frac{b}{m}]$, ${|f^{(n)}(x)|}^q$ for $q>1$ is $(\alpha ,m)$-convex on $[0,\frac{b}{m}]$, and $nq\ge p\ge 0$, then

(3.4)

Proof When $a<t<b$, by Lemma 2.1 and Hölder’s integral inequality, we have

(3.5)

where

$${\int }_a^t{(x-a)}^{(nq-p)/(q-1)}\mathrm{d}x=\frac{q-1}{nq+q-p-1}{(t-a)}^{(nq+q-p-1)/(q-1)}$$

(3.6)

and

$${\int }_t^b{(b-x)}^{(nq-p)/(q-1)}\mathrm{d}x=\frac{q-1}{nq+q-p-1}{(b-t)}^{(nq+q-p-1)/(q-1)}.$$

(3.7)

Since ${|f^{(n)}(x)|}^q$ is $(\alpha ,m)$-convex on $[0,\frac{b}{m}]$, we have

$$\begin{array}{c}{\int }_a^t{(x-a)}^p{|f^{(n)}(x)|}^q\mathrm{d}x\hfill \\ \le {\int }_a^t{(x-a)}^p\{{\left(\frac{t-x}{t-a}\right)}^{\alpha }{|f^{(n)}(a)|}^q+m[1-{\left(\frac{t-x}{t-a}\right)}^{\alpha }]{\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q\}\mathrm{d}x\hfill \\ =\frac{{(t-a)}^{p+1}}{p+1}[\alpha B(p+2,\alpha ){|f^{(n)}(a)|}^q+m(1-\alpha B(p+2,\alpha )){\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q]\hfill \end{array}$$

and

$$\begin{array}{c}{\int }_t^b{(b-x)}^p{|f^{(n)}(x)|}^q\mathrm{d}x\hfill \\ \le {\int }_t^b{(b-x)}^p\{{\left(\frac{b-x}{b-t}\right)}^{\alpha }{|f^{(n)}(t)|}^q+m[1-{\left(\frac{b-x}{b-t}\right)}^{\alpha }]{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q\}\mathrm{d}x\hfill \\ =\frac{{(b-t)}^{p+1}}{(p+1)(p+\alpha +1)}[(p+1){|f^{(n)}(t)|}^q+\alpha m{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q].\hfill \end{array}$$

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

$$\begin{array}{c}|\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)|\hfill \\ \le \frac{1}{(b-a)n!}{\left(\frac{q-1}{nq+q-p-1}\right)}^{1-1/q}{\left[\frac{1}{(p+1)(p+2)}\right]}^{1/q}\hfill \\ \times \{{(t-a)}^{n+1}{[{|f^{(n)}(a)|}^q+m(p+1){\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q]}^{1/q}\hfill \\ +{(b-t)}^{n+1}{\left[((p+1){|f^{(n)}(t)|}^q+m{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q)\right]}^{1/q}\};\hfill \end{array}$$

(2) if $m=1$, then

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

(3) if $m=\alpha =1$, we have

$$\begin{array}{c}|\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)|\hfill \\ \le \frac{1}{(b-a)n!}{\left(\frac{q-1}{nq+q-p-1}\right)}^{1-1/q}{\left[\frac{1}{(p+1)(p+2)}\right]}^{1/q}\left\{{(t-a)}^{n+1}\right[{|f^{(n)}(a)|}^q\hfill \\ +{(p+1){|f^{(n)}(t)|}^q]}^{1/q}+{(b-t)}^{n+1}{[(p+1){|f^{(n)}(t)|}^q+{|f^{(n)}(b)|}^q]}^{1/q}\}.\hfill \end{array}$$

Corollary 3.4 Under the conditions of Theorem  3.2,

(1) if $t=a$, then

(3.8)

(2) if $t=\frac{a+b}{2}$, then

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

(3) if $t=b$, then

Corollary 3.5 Under the conditions of Theorem  3.2,

(1) if $p=0$, then

$$\begin{array}{c}|\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)|\hfill \\ \le \frac{1}{(b-a)n!}{\left(\frac{q-1}{nq+q-1}\right)}^{1-1/q}{\left[\frac{1}{(\alpha +1)(\alpha +2)}\right]}^{1/q}\left\{{(t-a)}^{n+1}\right[{|f^{(n)}(a)|}^q\hfill \\ +{\alpha m{\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q]}^{1/q}+{(b-t)}^{n+1}{[{|f^{(n)}(t)|}^q+\alpha m{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q]}^{1/q}\};\hfill \end{array}$$

(2) if $p=q$, then

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

(3) if $p=nq$, then

$$\begin{array}{c}|\frac{1}{b-a}{\int }_a^{b}f(x)\mathrm{d}x-\frac{1}{b-a}\sum _{k=0}^{n-1}\frac{{(b-t)}^{k+1}+{(-1)}^k{(t-a)}^{k+1}}{(k+1)!}{f}^{(k)}(t)|\hfill \\ \le \frac{1}{(b-a)n!}{\left(\frac{1}{nq+1}\right)}^{1/q}\{{(t-a)}^{n+1}\hfill \\ \times {[\alpha B(nq+2,\alpha ){|f^{(n)}(a)|}^q+m(1-\alpha B(nq+2,\alpha )){\left|f^{(n)}\left(\frac{t}{m}\right)\right|}^q]}^{1/q}\hfill \\ +{(b-t)}^{n+1}{\left[\frac{1}{nq+\alpha +1}((nq+1){|f^{(n)}(t)|}^q+\alpha m{\left|f^{(n)}\left(\frac{b}{m}\right)\right|}^q)\right]}^{1/q}\}.\hfill \end{array}$$