Open Access

On Hadamard-Type Inequalities Involving Several Kinds of Convexity

Journal of Inequalities and Applications20102010:286845

DOI: 10.1155/2010/286845

Received: 14 May 2010

Accepted: 23 August 2010

Published: 26 August 2010

Abstract

We do not only give the extensions of the results given by Gill et al. (1997) for log-convex functions but also obtain some new Hadamard-type inequalities for log-convex https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq1_HTML.gif -convex, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq2_HTML.gif -convex functions.

1. Introduction

The following inequality is well known in the literature as Hadamard's inequality:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq3_HTML.gif is a convex function on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq4_HTML.gif of real numbers and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq5_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq6_HTML.gif This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, note worthy extension, generalizations, and numerous applications on this inequality; see ([16]) where further references are given.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq7_HTML.gif be on interval in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq8_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq9_HTML.gif is said to be convex if, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq11_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ2_HTML.gif
(1.2)

(see [5], Page 1). Geometrically, this means that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq13_HTML.gif are three distinct points on the graph of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq14_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq15_HTML.gif between https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq17_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq18_HTML.gif is on or below chord https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq19_HTML.gif

Recall that a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq20_HTML.gif is said to be log-convex function if, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq22_HTML.gif , one has the inequality (see [5], Page 3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ3_HTML.gif
(1.3)

It is said to be log-concave if the inequality in (1.3) is reversed.

In [7], Toader defined https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq23_HTML.gif -convexity as follows.

Definition 1.1.

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq25_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq26_HTML.gif -convex, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq27_HTML.gif , if one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ4_HTML.gif
(1.4)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq29_HTML.gif We say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq30_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq31_HTML.gif -concave if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq32_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq33_HTML.gif -convex.

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq34_HTML.gif the class of all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq35_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq36_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq37_HTML.gif (if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq38_HTML.gif ). Obviously, if we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq39_HTML.gif Definition 1.1 recaptures the concept of standard convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq40_HTML.gif

In [8], Miheşan defined https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq41_HTML.gif -convexity as in the following:

Definition 1.2.

The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq43_HTML.gif , is said to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq44_HTML.gif -convex, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq45_HTML.gif , if one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ5_HTML.gif
(1.5)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq47_HTML.gif .

Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq48_HTML.gif the class of all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq49_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq50_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq51_HTML.gif . It can be easily seen that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq52_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq53_HTML.gif -convexity reduces to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq54_HTML.gif -convexity and for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq56_HTML.gif -convexity reduces to the concept of usual convexity defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq57_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq58_HTML.gif .

For recent results and generalizations concerning https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq59_HTML.gif -convex and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq60_HTML.gif -convex functions, see ([912]).

In the literature, the logarithmic mean of the positive real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq61_HTML.gif is defined as the following:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ6_HTML.gif
(1.6)

(for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq62_HTML.gif , we put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq63_HTML.gif ).

In [13], Gill et al. established the following results.

Theorem 1.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq64_HTML.gif be a positive, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq65_HTML.gif -convex function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq66_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq67_HTML.gif is a logarithmic mean of the positive real numbers as in (1.6).

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq68_HTML.gif a positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq69_HTML.gif -concave function, the inequality is reversed.

Corollary 1.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq70_HTML.gif be positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq71_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq72_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ8_HTML.gif
(1.8)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq73_HTML.gif is a positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq74_HTML.gif -concave function, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ9_HTML.gif
(1.9)

For some recent results related to the Hadamard's inequalities involving two https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq75_HTML.gif -convex functions, see [14] and the references cited therein. The main purpose of this paper is to establish the general version of inequalities (1.7) and new Hadamard-type inequalities involving two https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq76_HTML.gif -convex functions, two https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq77_HTML.gif -convex functions, or two https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq78_HTML.gif -convex functions using elementary analysis.

2. Main Results

We start with the following theorem.

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq79_HTML.gif    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq80_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq81_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq83_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq84_HTML.gif . Then the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ10_HTML.gif
(2.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq85_HTML.gif is a logarithmic mean of positive real numbers.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq86_HTML.gif a positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq87_HTML.gif -concave function, the inequality is reversed.

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq88_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq89_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq90_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq91_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ11_HTML.gif
(2.2)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq92_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq93_HTML.gif Writing (2.2) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq94_HTML.gif and multiplying the resulting inequalities, it is easy to observe that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ12_HTML.gif
(2.3)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq95_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq96_HTML.gif

Integrating inequality (2.3) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq97_HTML.gif over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq98_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ13_HTML.gif
(2.4)
As
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ14_HTML.gif
(2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ15_HTML.gif
(2.6)

the theorem is proved.

Remark 2.2.

By taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq99_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq100_HTML.gif in Theorem 2.1 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq101_HTML.gif we obtain (1.7).

Corollary 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq102_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq103_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq104_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq105_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq106_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ16_HTML.gif
(2.7)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq107_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq108_HTML.gif are positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq109_HTML.gif -concave functions, then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ17_HTML.gif
(2.8)

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq110_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq111_HTML.gif be positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq112_HTML.gif -convex functions. Then by Theorem 2.1 we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ18_HTML.gif
(2.9)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq113_HTML.gif , whence (2.7). Similarly we can prove (2.8).

Remark 2.4.

By taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq114_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq115_HTML.gif in (2.7) and (2.8) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq116_HTML.gif we obtain the inequalities of Corollary 1.4.

We will now point out some new results of the Hadamard type for log-convex, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq117_HTML.gif -convex, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq118_HTML.gif -convex functions, respectively.

Theorem 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq119_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq120_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq121_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq122_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq123_HTML.gif Then the following inequalities hold:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ19_HTML.gif
(2.10)

Proof.

We can write
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ20_HTML.gif
(2.11)
Using the elementary inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq124_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq125_HTML.gif reals) and equality (2.11), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ21_HTML.gif
(2.12)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq126_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq127_HTML.gif -convex functions, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ22_HTML.gif
(2.13)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq128_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq129_HTML.gif .

Rewriting (2.12) and (2.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ23_HTML.gif
(2.14)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ24_HTML.gif
(2.15)
Integrating both sides of (2.14) and (2.15) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq130_HTML.gif over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq131_HTML.gif , respectively, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ25_HTML.gif
(2.16)

Combining (2.16), we get the desired inequalities (2.10). The proof is complete.

Theorem 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq132_HTML.gif be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq133_HTML.gif -convex functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq135_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq136_HTML.gif Then the following inequalities hold:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ26_HTML.gif
(2.17)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq137_HTML.gif is a logarithmic mean of positive real numbers.

Proof.

From inequality (2.14), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ27_HTML.gif
(2.18)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq138_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq139_HTML.gif

Using the elementary inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq140_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq141_HTML.gif reals) on the right side of the above inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ28_HTML.gif
(2.19)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq142_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq143_HTML.gif -convex functions, then we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ29_HTML.gif
(2.20)
Integrating both sides of (2.19) and (2.20) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq144_HTML.gif over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq145_HTML.gif , respectively, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ30_HTML.gif
(2.21)

Combining (2.21), we get the required inequalities (2.17). The proof is complete.

Theorem 2.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq146_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq147_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq148_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq149_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq150_HTML.gif is nonincreasing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq151_HTML.gif -convex function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq152_HTML.gif is nonincreasing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq153_HTML.gif -convex function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq154_HTML.gif for some fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq155_HTML.gif then the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ31_HTML.gif
(2.22)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ32_HTML.gif
(2.23)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ33_HTML.gif
(2.24)

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq156_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq157_HTML.gif -convex function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq158_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq159_HTML.gif -convex function, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ34_HTML.gif
(2.25)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq160_HTML.gif . It is easy to observe that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ35_HTML.gif
(2.26)
Using the elementary inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq161_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq162_HTML.gif reals), (2.25) on the right side of (2.26) and making the charge of variable and since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq163_HTML.gif is nonincreasing, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ36_HTML.gif
(2.27)
Analogously we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ37_HTML.gif
(2.28)

Rewriting (2.27) and (2.28), we get the required inequality in (2.22). The proof is complete.

Theorem 2.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq164_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq165_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq166_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq167_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq168_HTML.gif is nonincreasing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq169_HTML.gif -convex function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq170_HTML.gif is nonincreasing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq171_HTML.gif -convex function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq172_HTML.gif for some fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq173_HTML.gif Then the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ38_HTML.gif
(2.29)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ39_HTML.gif
(2.30)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ40_HTML.gif
(2.31)

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq174_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq175_HTML.gif -convex function and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq176_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq177_HTML.gif -convex function, then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ41_HTML.gif
(2.32)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq178_HTML.gif . It is easy to observe that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ42_HTML.gif
(2.33)
Using the elementary inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq179_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq180_HTML.gif reals), (2.32) on the right side of (2.33) and making the charge of variable and since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq181_HTML.gif is nonincreasing, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ43_HTML.gif
(2.34)
Analogously we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_Equ44_HTML.gif
(2.35)

Rewriting (2.34) and (2.35), we get the required inequality in (2.29). The proof is complete.

Remark 2.9.

In Theorem 2.8, if we choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F286845/MediaObjects/13660_2010_Article_2108_IEq182_HTML.gif , we obtain the inequality of Theorem 2.7.

Authors’ Affiliations

(1)
Department of Mathematics, K.K. Education Faculty, Atatürk University
(2)
Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University
(3)
School of Computational and Applied Mathematics, University of the Witwatersrand

References

  1. Alomari M, Darus M: On the Hadamard's inequality for log-convex functions on the coordinates. Journal of Inequalities and Applications 2009, 2009:-13.Google Scholar
  2. Zhang X-M, Chu Y-M, Zhang X-H: The Hermite-Hadamard type inequality of GA-convex functions and its applications. Journal of Inequalities and Applications 2010, 2010:-11.Google Scholar
  3. Dinu C: Hermite-Hadamard inequality on time scales. Journal of Inequalities and Applications 2008, 2008:-24.Google Scholar
  4. Dragomir SS, Pearce CEM: Selected Topics on Hermite-Hadamard Inequalities and Applications. RGMIA Monographs, Victoria University, 2000, http://www.staff.vu.edu.au/rgmia/monographs/hermite_hadamard.html RGMIA Monographs, Victoria University, 2000,
  5. Mitrinović DS, Pečarić JE, Fink AM: Classical and New Inequalities in Analysis, Mathematics and Its Applications. Volume 61. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xviii+740.View ArticleGoogle Scholar
  6. Set E, Özdemir ME, Dragomir SS: On the Hermite-Hadamard inequality and other integral inequalities involving two functions. Journal of Inequalities and Applications 2010, -9.Google Scholar
  7. Toader G: Some generalizations of the convexity. In Proceedings of the Colloquium on Approximation and Optimization, Cluj-Napoca, Romania. University of Cluj-Napoca; 1985:329–338.Google Scholar
  8. Miheşan VG: A generalization of the convexity. Proceedings of the Seminar on Functional Equations, Approximation and Convexity, 1993, Cluj-Napoca, RomaniaGoogle Scholar
  9. Bakula MK, Özdemir ME, Pečarić J: Hadamard type inequalities for m -convex and -convex functions. Journal of Inequalities in Pure and Applied Mathematics 2008., 9, article no. 96:Google Scholar
  10. Bakula MK, Pečarić J, Ribičić M: Companion inequalities to Jensen's inequality for m -convex and ( α , m )-convex functions. Journal of Inequalities in Pure and Applied Mathematics 2006., 7(5, article no. 194):
  11. Pycia M: A direct proof of the s -Hölder continuity of Breckner s -convex functions. Aequationes Mathematicae 2001, 61(1–2):128–130. 10.1007/s000100050165MathSciNetView ArticleMATHGoogle Scholar
  12. Özdemir ME, Avci M, Set E: On some inequalities of Hermite-Hadamard type via m-convexity. Applied Mathematics Letters 2010, 23(9):1065–1070. 10.1016/j.aml.2010.04.037MathSciNetView ArticleMATHGoogle Scholar
  13. Gill PM, Pearce CEM, Pečarić J: Hadamard's inequality for r -convex functions. Journal of Mathematical Analysis and Applications 1997, 215(2):461–470. 10.1006/jmaa.1997.5645MathSciNetView ArticleMATHGoogle Scholar
  14. Pachpatte BG: A note on integral inequalities involving two log-convex functions. Mathematical Inequalities & Applications 2004, 7(4):511–515.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Erhan Set et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.