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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)
We established a new Hermit-Hadamard type inequality for GA-convex functions. As applications, we obtain two new Gautschi type inequalities for gamma function.
Let be a convex (concave) function on ; the well-known Hermite-Hadamard's inequality  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 , 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.
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  first proved that
A strengthened upper bound was given by Erber :
In , Kečkić and Vasić established the following double inequality for :
In , Kershaw obtained
for and .
In , Zhang and Chu proved
for all .
In , 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.
If , then
If , then
In order to establish our main results we need several lemmas, which we present in this section.
Simple computations lead to
Lemma 2.2 (see [28, Lemma ]).
If , then
where , , , , , , .
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 .
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
is GA-concave on .
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.
is GA-convex on .
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.
. Taking in (2.2) and in (2.3) we get
Inequalities (2.11) and (2.12) together with lead to
. 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
. 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).
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.
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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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
- Special Function
- Simple Computation
- Gamma Function
- Elementary Computation
- Intensive Research