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# The Hermite-Hadamard Type Inequality of GA-Convex Functions and Its Application

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 507560 (2010)

## Abstract

We established a new Hermit-Hadamard type inequality for GA-convex functions. As applications, we obtain two new Gautschi type inequalities for gamma function.

## 1. Introduction

Let be a convex (concave) function on ; the well-known Hermite-Hadamard's inequality [1] can be expressed as

Recently, Hermite-Hadamard's inequality has been the subject of intensive research. In particular, many improvements, generalizations, and applications for the Hermite-Hadamard's inequality can be found in the literature [2–20].

Let be an interval; a real-valued function is said to be GA-convex (concave) on if for all and .

In [21], Anderson et al. discussed the GA and related kinds of convexity; some applications to special functions were presented.

For , let , , and be the geometric, logarithmic, identric, and arithmetic means of and , respectively. Then

The first purpose of this paper is to establish the following new Hermite-Hadamard type inequality for GA-convex (concave) functions.

Theorem 1.1.

If and is a differentiable GA-convex (concave) function, then

For real and positive values of , the Euler gamma function and its logarithmic derivative , the so-called digamma function, are defined by

The ratio has attracted the attention of many mathematicians and physicists. Gautschi [22] first proved that

for and

A strengthened upper bound was given by Erber [23]:

In [24], Kečkić and Vasić established the following double inequality for :

In [25], Kershaw obtained

for and .

In [26], Zhang and Chu proved

for all .

In [27], Zhang and Chu presented

for all .

The second purpose of this paper is to establish the following two new Gautschi type inequalities by using Theorem 1.1.

Theorem 1.2.

If , then

Theorem 1.3.

If , then

## 2. Lemmas

In order to establish our main results we need several lemmas, which we present in this section.

Lemma 2.1.

One has

Proof.

Simple computations lead to

Lemma 2.2 (see [28, Lemma ]).

If , then

where , , , , , , .

Lemma 2.3.

Suppose that is an interval and is a real-valued function. If is second-order differentiable on , then is GA-convex (concave) on if and only if

for all .

Proof.

Lemma 2.3 follows easily from the basic properties of convex (concave) functions and the fact that is GA-convex (concave) on if and only if is convex (concave) on .

Lemma 2.4 (see [29, Theorem ]).

If , then

Lemma 2.5.

is GA-concave on .

Proof.

Differentiating the well-known identity we get

From inequalities (2.5) and (2.6) we have

Inequality (2.7) leads to

Therefore, Lemma 2.5 follows from (2.8) and Lemma 2.3.

Lemma 2.6.

is GA-convex on .

Proof.

Simple computation leads to

From (2.9) and Lemma 2.3 we know that we need only to prove that

We divide the proof into three cases.

Case 1.

. Taking in (2.2) and in (2.3) we get

Inequalities (2.11) and (2.12) together with lead to

Case 2.

. It is well-known that

where is Euler's constant.

Differentiating (2.14) we get

We clearly see that is increasing in for ; hence (2.15) and (2.16) lead to

It follows from inequality (2.17), Lemma 2.1, and that

Case 3.

. Since is decreasing in for , hence (2.15) and (2.16) imply that

From (2.19), Lemma 2.1, and we get

It is not difficult to verify that

Therefore, inequality (2.10) follows from (2.20) and (2.21).

## 3. Proof of Theorems 1.1, 1.2, and 1.3

Proof of Theorem 1.1.

Suppose that is a GA-convex function. For any fixed , if , then is convex on and

Inequality (3.1) implies that

Let , then inequality (3.2) leads to that for . Hence , namely,

Using a similar method we get

Let , then

From inequalities (3.3) and (3.4) together with (3.5) we clearly see that

Next for any , let , then and . From the definition of GA-convex function and the transformation to variable of integration we get

Therefore, Theorem 1.1 follows from inequalities (3.6) and (3.7).

Proof of Theorem 1.2.

From Lemmas 2.5 and 2.6 together with Theorem 1.1 we clearly see that

Therefore, Theorem 1.2 follows from (3.8) and (3.9).

Proof of Theorem 1.3.

From Lemmas 2.5 and 2.6 together with Theorem 1.1 we get

Inequalities (3.10) and (3.11) lead to

Therefore, Theorem 1.3 follows from (3.12) and (3.13).

Remark 3.1.

Making use of a computer and the mathematica software we can show that the bounds in Theorems 1.2 and 1.3 are stronger than that in inequalities (1.9) and (1.10) for some and . In fact, if we let , , , , , and , then we have Tables 1 and 2 via elementary computation.

Remark 3.2.

We clear see that the lower bound in Theorem 1.3 is stronger than that in inequality (1.9) for all .

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## Acknowledgments

The authors wish to thank the anonymous referee for their very careful reading of the manuscript and fruitful comments and suggestions. This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y607128.

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Zhang, XM., Chu, YM. & Zhang, XH. The Hermite-Hadamard Type Inequality of GA-Convex Functions and Its Application.
*J Inequal Appl* **2010**, 507560 (2010). https://doi.org/10.1155/2010/507560

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DOI: https://doi.org/10.1155/2010/507560