# A Class of Logarithmically Completely Monotonic Functions Associated with a Gamma Function

- Tie-Hong Zhao
^{1}Email author and - Yu-Ming Chu
^{1}

**2010**:392431

https://doi.org/10.1155/2010/392431

© T.-H. Zhao and Y.-M. Chu. 2010

**Received: **15 November 2010

**Accepted: **27 December 2010

**Published: **30 December 2010

## Abstract

## 1. Introduction

For extension of these functions to complex variables and for basic properties, see [1]. These functions play central roles in the theory of special functions and have lots of extensive applications in many branches, for example, statistics, physics, engineering, and other mathematical sciences. Over the past half century monotonicity properties of these functions have attracted the attention of many authors (see [2–22]).

for all and . Moreover, is said to be strictly completely monotonic if inequality (1.2) is strict.

for all and . Moreover, is said to be strictly logarithmically completely monotonic if inequality (1.3) is strict.

Recently, the completely monotonic or logarithmically completely monotonic functions have been the subject of intensive research. In particular, many remarkable results for the complete monotonicity or logarithmically complete monotonicity involving the gamma, psi and polygamma functions can be found in the literature [18, 19, 23–42].

for or and , and inequality (1.5) is reversed for and .

for with and . In order to establish the best bounds in Kershaw's inequality (1.4), the following monotonicity and convexity properties of are established in [13, 44, 45]: the function is either convex and decreasing for or concave and increasing for .

Then, the function is strictly logarithmically completely monotonic on if and only if . So is the function if and only if .

Our main results are summarized as follows.

Theorem 1.1.

Let , , and is defined as (1.8), then

(1) is strictly logarithmically completely monotonic on if and only if ;

(2) is strictly logarithmically completely monotonic on if and only if .

As applications of Theorem 1.1, one has the following corollaries.

Corollary 1.2.

Corollary 1.3.

## 2. Lemmas

In order to prove our Theorem 1.1, we need serval lemmas which we collect in this section. In our second lemma we present the area of to determine positive (or negative) for a function, which plays a crucial role in the proof of our result Theorem 1.1 given in Section 3.

Obviously, if , then . It follows from that .

It follows from the properties of the quadratic equation that for and for or .

By (2.5) we know that the minimal value of can be attained at , that is . Moreover, is strictly decreasing on and strictly increasing on .

as . In other words, and has the asymptotic line .

Lemma 2.1.

for and , where is Euler's constant.

Lemma 2.2.

Let and . Then the following statements are true:

(3)if and , then there exist such that for and for .

Proof.

Case 1.

Case 2.

Therefore, there exists such that for and for follows from (2.17), which implies that is strictly decreasing on and strictly increasing on . It follows from (2.12) and that there exists such that for and for . By the same argument, it follows from (2.10) and that there exists such that for and for .

Therefore, for follows from (2.9).

- (2)
If , then from Figure 1 we know that could be positive or negative. We divide the proof into two cases.

Case 1.

Case 2.

If , then from (2.13) and (2.15) we know that there exists such that for and for . Thus is strictly decreasing on and strictly increasing on . It follows from (2.12) and that there exists such that for and for . By the same argument as Case 1, for follows from (2.9) and (2.10).

We state a simple lemma as the results of [12, 47].

Lemma 2.3.

## 3. Proof of Theorem 1.1

Proof of Theorem 1.1.

holds for . Therefore, is strictly logarithmically completely monotonic on that follows from (3.4) and (3.5).

- (2)

Therefore, is strictly logarithmically completely monotonic on that follows from (3.6), (3.8), and Lemma 2.2(2).

Conversely, if and , then we can divide the set into two subsets: and . Therefore, it follows from Lemma 2.2(1) and (3) that is not strictly logarithmically completely monotonic on for .

Remark 1.

Furthermore, the advantage of our inequalities is to give the upper and lower bounds of Kershaw's inequality for and while Laforgia established only one side of Kershaw's inequality.

## Declarations

### Acknowledgment

This work was supported by the National Science Foundation of China under Grant no. 11071069 and the Natural Science Foundation of Zhejiang Province under Grant no. Y7080106.

## Authors’ Affiliations

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