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

Shu-Ping Bai1, Shu-Hong Wang1 and Feng Qi23*

Author 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

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Journal of Inequalities and Applications 2012, 2012:267  doi:10.1186/1029-242X-2012-267


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/267


Received:12 June 2012
Accepted:7 November 2012
Published:22 November 2012

© 2012 Bai et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex.

MSC: 26D15, 26A51, 41A55.

Keywords:
Hermite-Hadamard’s integral inequality; differentiable function; <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex function

1 Introduction

Throughout this paper, we adopt the following notations:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M4">View MathML</a>

(1.1)

We recall some definitions of several convex functions.

Definition 1.1 A function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M5">View MathML</a> is said to be convex if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M6">View MathML</a>

(1.2)

holds for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M7">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M8">View MathML</a>.

Definition 1.2 ([1])

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M9">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M10">View MathML</a>, if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M11">View MathML</a>

(1.3)

is valid for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M12">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M8">View MathML</a>, then we say that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M14">View MathML</a> is an m-convex function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M15">View MathML</a>.

Definition 1.3 ([2])

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M9">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M17">View MathML</a>, if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M18">View MathML</a>

(1.4)

is valid for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M12">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M8">View MathML</a>, then we say that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M14">View MathML</a> is an <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M22">View MathML</a>-convex function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M15">View MathML</a>.

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<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M24">View MathML</a>be a differentiable mapping and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M25">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M26">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M27">View MathML</a>is convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M29">View MathML</a>

(1.5)

Theorem 1.2 ([[4], Theorem 2])

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M30">View MathML</a>bem-convex and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M10">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M32">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M34">View MathML</a>

(1.6)

Theorem 1.3 ([[2], Theorem 2.2])

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M35">View MathML</a>be an open interval and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M36">View MathML</a>be a differentiable function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M37">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M39">View MathML</a>ism-convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28">View MathML</a>for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M41">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M42">View MathML</a>, then

(1.7)

Theorem 1.4 ([[2], Theorem 3.1])

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M35">View MathML</a>be an open interval and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M36">View MathML</a>be a differentiable function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M37">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M48">View MathML</a>is<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28">View MathML</a>for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M17">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M42">View MathML</a>, then

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M54">View MathML</a>

(1.8)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M55">View MathML</a>

(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 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M56">View MathML</a>-convex.

2 A lemma

In order to find inequalities of Hermite-Hadamard type for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M56">View MathML</a>-convex functions, we need the following lemma.

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

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M58">View MathML</a>be ann-time differentiable function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M59">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M60">View MathML</a>is absolutely continuous on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M28">View MathML</a>. Then the identity

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M62">View MathML</a>

(2.1)

holds for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M63">View MathML</a>, where the kernel<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M64">View MathML</a>is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M65">View MathML</a>

(2.2)

3 Hermite-Hadamard type inequalities for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex functions

We now set off to establish some new integral inequalities of Hermite-Hadamard type for n-time differentiable <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex functions.

Theorem 3.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M30">View MathML</a>be ann-time differentiable function for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M69">View MathML</a>and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M33">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M71">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M72">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M73">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M42">View MathML</a>is<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M76">View MathML</a>, then

(3.1)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M63">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M79">View MathML</a>is the beta function

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M80">View MathML</a>

(3.2)

Proof If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M81">View MathML</a>, by Lemma 2.1, Hölder’s integral inequality, and the <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M83">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M84">View MathML</a>

Substituting

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M85">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M86">View MathML</a>

into the above inequality leads to the inequality (3.1) for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M87">View MathML</a>.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89">View MathML</a>, by virtue of Lemma 2.1 and the property that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M90">View MathML</a> is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M76">View MathML</a>, we have

and

The inequality (3.1) for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89">View MathML</a> follows. Theorem 3.1 is thus proved. □

Corollary 3.1Under the conditions of Theorem 3.1,

(1) when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M97">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M98">View MathML</a>

(2) when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M99">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M100">View MathML</a>

(3) when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M101">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M102">View MathML</a>

(4) when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M103">View MathML</a>, we have

Corollary 3.2Under the conditions of Theorem 3.1,

(1) when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88">View MathML</a>, we have

(3.3)

(2) when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M107">View MathML</a>, we have

(3) when<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89">View MathML</a>, we have

Theorem 3.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M63">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M30">View MathML</a>be ann-time differentiable function for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M69">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M114">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M17">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M72">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M73">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M118">View MathML</a>is<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M76">View MathML</a>, and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M121">View MathML</a>, then

(3.4)

Proof When <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M81">View MathML</a>, by Lemma 2.1 and Hölder’s integral inequality, we have

(3.5)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M125">View MathML</a>

(3.6)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M126">View MathML</a>

(3.7)

Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M90">View MathML</a> is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M1">View MathML</a>-convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M129">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M130">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M131">View MathML</a>

Hence, the inequality (3.4) follows.

When <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89">View MathML</a>, the proof of the inequality (3.4) is similar to the above argument. The proof of Theorem 3.2 is complete. □

Corollary 3.3Under the conditions of Theorem 3.2,

(1) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M99">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M135">View MathML</a>

(2) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M101">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M137">View MathML</a>

(3) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M138">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M139">View MathML</a>

Corollary 3.4Under the conditions of Theorem 3.2,

(1) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M88">View MathML</a>, then

(3.8)

(2) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M107">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M143">View MathML</a>

(3) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M89">View MathML</a>, then

Corollary 3.5Under the conditions of Theorem 3.2,

(1) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M146">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M147">View MathML</a>

(2) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M148">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M149">View MathML</a>

(3) if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M150">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/267/mathml/M151">View MathML</a>

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and read and approved the final manuscript.

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.

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