Open Access

On the Hermite-Hadamard Inequality and Other Integral Inequalities Involving Two Functions

Journal of Inequalities and Applications20102010:148102

DOI: 10.1155/2010/148102

Received: 25 September 2009

Accepted: 31 March 2010

Published: 24 May 2010

Abstract

We establish some new Hermite-Hadamard-type inequalities involving product of two functions. Other integral inequalities for two functions are obtained as well. The analysis used in the proofs is fairly elementary and based on the use of the Minkowski, Hölder, and Young inequalities.

1. Introduction

Integral inequalities have played an important role in the development of all branches of Mathematics.

In [1, 2], Pachpatte established some Hermite-Hadamard-type inequalities involving two convex and log-convex functions, respectively. In [3], Bakula et al. improved Hermite-Hadamard type inequalities for products of two https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq1_HTML.gif -convex and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq2_HTML.gif -convex functions. In [4], analogous results for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq3_HTML.gif -convex functions were proved by Kirmaci et al.. General companion inequalities related to Jensen's inequality for the classes of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq4_HTML.gif -convex and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq5_HTML.gif -convex functions were presented by Bakula et al. (see [5]).

For several recent results concerning these types of inequalities, see [612] where further references are listed.

The aim of this paper is to establish several new integral inequalities for nonnegative and integrable functions that are related to the Hermite-Hadamard result. Other integral inequalities for two functions are also established.

In order to prove some inequalities related to the products of two functions we need the following inequalities. One of inequalities of this type is the following one.

Barnes-Gudunova-Levin Inequality (see [1315] and references therein)

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq7_HTML.gif be nonnegative concave functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq8_HTML.gif . Then, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq9_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ1_HTML.gif
(1.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ2_HTML.gif
(1.2)
In the special case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq10_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ3_HTML.gif
(1.3)
with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ4_HTML.gif
(1.4)

To prove our main results we recall some concepts and definitions.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq11_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq12_HTML.gif be two positive https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq13_HTML.gif -tuples, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq14_HTML.gif . Then, on putting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq15_HTML.gif the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq16_HTML.gif th power mean of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq17_HTML.gif with weights https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq18_HTML.gif is defined [16] by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ5_HTML.gif
(1.5)
Note that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq19_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ6_HTML.gif
(1.6)

(see, e.g., [10, page 15]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq21_HTML.gif . The https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq22_HTML.gif -norm of the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq23_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq24_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ7_HTML.gif
(1.7)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq25_HTML.gif is the set of all functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq26_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq27_HTML.gif .

One can rewrite the inequality (1.1) as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ8_HTML.gif
(1.8)

For several recent results concerning https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq28_HTML.gif -norms we refer the interested reader to [17].

Also, we need some important inequalities.

Minkowski Integral Inequality (see page 1 in [18])

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq29_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq30_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ9_HTML.gif
(1.9)

Hermite-Hadamard's Inequality (see page 10 in [10])

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq31_HTML.gif be a convex function on interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq32_HTML.gif of real numbers and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq33_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq34_HTML.gif . Then the following Hermite-Hadamard inequality for convex functions holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ10_HTML.gif
(1.10)
If the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq35_HTML.gif is concave, the inequality (1.10) can be written as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ11_HTML.gif
(1.11)

For recent results, refinements, counterparts, generalizations, and new Hermite-Hadamard-type inequalities, see [1921].

A Reversed Minkowski Integral Inequality (see page 2 in [18])

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq37_HTML.gif be positive functions satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ12_HTML.gif
(1.12)
Then, putting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq38_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ13_HTML.gif
(1.13)

One of the most important inequalities of analysis is Hölder's integral inequality which is stated as follows (for its variant see [10, page 106]).

Hölder Integral Inequality

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq40_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq41_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq42_HTML.gif are real functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq43_HTML.gif and if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq44_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq45_HTML.gif are integrable functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq46_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ14_HTML.gif
(1.14)

with equality holding if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq47_HTML.gif almost everywhere, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq48_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq49_HTML.gif are constants.

Remark 1.1.

Observe that whenever, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq50_HTML.gif is concave on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq51_HTML.gif the nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq52_HTML.gif is also concave on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq53_HTML.gif . Namely,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ15_HTML.gif
(1.15)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ16_HTML.gif
(1.16)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq54_HTML.gif ; using the power-mean inequality (1.6), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ17_HTML.gif
(1.17)

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq55_HTML.gif , similarly if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq56_HTML.gif is concave on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq57_HTML.gif the nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq58_HTML.gif is concave on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq59_HTML.gif .

2. The Results

Theorem 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq60_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq62_HTML.gif , be nonnegative functions such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq63_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq64_HTML.gif are concave on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq65_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ18_HTML.gif
(2.1)
and if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq66_HTML.gif , then one has
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ19_HTML.gif
(2.2)

Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq67_HTML.gif is the Barnes-Gudunova-Levin constant given by (1.1).

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq68_HTML.gif are concave functions on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq69_HTML.gif , then from (1.11) and Remark 1.1 we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ20_HTML.gif
(2.3)
By multiplying the above inequalities, we obtain (2.4) and (2.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ21_HTML.gif
(2.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ22_HTML.gif
(2.5)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq70_HTML.gif , then it easy to show that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ23_HTML.gif
(2.6)

Thus, by applying Barnes-Gudunova-Levin inequality to the right-hand side of (2.4) with (2.6), we get (2.1).

Applying the Hölder inequality to the left-hand side of (2.5) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq71_HTML.gif , we get (2.2).

Theorem 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq72_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq73_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq74_HTML.gif be positive functions with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ24_HTML.gif
(2.7)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ25_HTML.gif
(2.8)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq75_HTML.gif

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq76_HTML.gif are positive, as in the proof of the inequality (1.13) (see [18, page 2]), we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ26_HTML.gif
(2.9)
By multiplying the above inequalities, we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ27_HTML.gif
(2.10)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq78_HTML.gif by applying the Minkowski integral inequality to the right hand side of (2.10), we obtain inequality (2.8).

Theorem 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq79_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq80_HTML.gif be as in Theorem 2.1. Then the following inequality holds:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ28_HTML.gif
(2.11)

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq81_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq82_HTML.gif are concave on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq83_HTML.gif , then from (1.11) we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ29_HTML.gif
(2.12)
which imply that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ30_HTML.gif
(2.13)
On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq84_HTML.gif from (1.6) we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ31_HTML.gif
(2.14)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ32_HTML.gif
(2.15)
which imply that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ33_HTML.gif
(2.16)
Combining (2.13) and (2.16), we obtain the desired inequality as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ34_HTML.gif
(2.17)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ35_HTML.gif
(2.18)
To prove the following theorem we need the following Young-type inequality (see [7, page 117]):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ36_HTML.gif
(2.19)

Theorem 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq85_HTML.gif be functions such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq86_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq87_HTML.gif are in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq88_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ37_HTML.gif
(2.20)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ38_HTML.gif
(2.21)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ39_HTML.gif
(2.22)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq89_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq90_HTML.gif

Proof.

From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq91_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ40_HTML.gif
(2.23)
From (2.19) with (2.23) we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ41_HTML.gif
(2.24)
Using the elementary inequality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq92_HTML.gif , ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_IEq94_HTML.gif ) in (2.24), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F148102/MediaObjects/13660_2009_Article_2063_Equ42_HTML.gif
(2.25)

This completes the proof of the inequality in (2.21).

Declarations

Acknowledgment

The authors thank the careful referees for some good advices which have improved the final version of this paper.

Authors’ Affiliations

(1)
Department of Mathematics, K. K. Education Faculty, Atatürk University
(2)
Graduate School of Natural and Applied Sciences, Ağrı İbrahim Çeçen University
(3)
Research Group in Mathematical Inequalities & Applications, School of Engineering & Science, Victoria University

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© Erhan Set et al. 2010

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