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

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

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

for and

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

(1.6)

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

(1.7)

In [25], Kershaw obtained

(1.8)

for and .

In [26], Zhang and Chu proved

(1.9)

for all .

In [27], Zhang and Chu presented

(1.10)

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

(1.11)

Theorem 1.3.

If , then

(1.12)

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

(2.1)

Lemma 2.2 (see [28, Lemma ]).

If , then

(2.2)

(2.3)

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

(2.4)

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

(2.5)

Lemma 2.5.

is GA-concave on .

Proof.

Differentiating the well-known identity we get

(2.6)

From inequalities (2.5) and (2.6) we have

(2.7)

Inequality (2.7) leads to

(2.8)

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

Lemma 2.6.

is GA-convex on .

Proof.

Simple computation leads to

(2.9)

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

(2.10)

We divide the proof into three cases.

Case 1.

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

(2.11)

(2.12)

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

(2.13)

Case 2.

. It is well-known that

(2.14)

where is Euler's constant.

Differentiating (2.14) we get

(2.15)

(2.16)

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

(2.17)

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

(2.18)

Case 3.

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

(2.19)

From (2.19), Lemma 2.1, and we get

(2.20)

It is not difficult to verify that

(2.21)

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

(3.1)

Inequality (3.1) implies that

(3.2)

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

(3.3)

Using a similar method we get

(3.4)

Let , then

(3.5)

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

(3.6)

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

(3.7)

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

(3.8)

(3.9)

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

(3.10)

(3.11)

Inequalities (3.10) and (3.11) lead to

(3.12)

(3.13)

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.

Table 1 Comparison of and with and for some and

Table 2 Comparison of and with for some and .

Remark 3.2.

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