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

Abstract

We show that the function is strictly logarithmically completely monotonic on if and only if and is strictly logarithmically completely monotonic on if and only if .

1. Introduction

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

(1.1)

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]).

Recall that a real-valued function is said to be completely monotonic on if has derivatives of all orders on and

(1.2)

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

Recall also that a positive real-valued function is said to be logarithmically completely monotonic on if has derivatives of all orders on and its logarithm satisfies

(1.3)

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].

The Kershaw's inequality in [21] states that the double inequality

(1.4)

holds for and . In [43], Laforgia extends the both sides of inequality in (1.4) as follows:

(1.5)

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

Let us define

(1.6)

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 .

This work is motivated by an paper of Guo [46], who proved that the function

(1.7)

is strictly logarithmically concave and strictly increasing from onto . It is natural to ask for an extension of this result: is logarithmically complete monotonic? We will give the positive answer. Actually, we investigate a more general problem. The goal of this article is to discuss the logarithmically complete monotonicity properties of the functions

(1.8)

on and for fixed .

Recently Chen et al. [38, Theorem 1] proved the following result: let and be real numbers, define for ,

(1.9)

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.

For and , one has the double inequalities for the ratio of the gamma functions

(1.10)

In particular, one has

(1.11)

for and , and

(1.12)

for and .

Corollary 1.3.

For and , one has the following double inequality

(1.13)

where .

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.

Let be a function defined on as

(2.1)

We will discuss the properties for this function and refer to view Figure 1 more clearly.

Figure 1

The shading area is denoted by . Otherwise, . The red curve is the graph of with an asymptotic line .

The function can be interpreted as a quadric equation with respect to , that is

(2.2)

where , , and its discriminant function

(2.3)

Obviously, if , then . It follows from that .

If , then . We can solve two roots of the equation , which are

(2.4)

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

Differentiating with respect to , one has

(2.5)

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 .

Obviously, is strictly increasing on . Note that

(2.6)

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

Lemma 2.1.

The psi or digamma function, the logarithmic derivative of the gamma function, and the polygamma functions can be expressed as

(2.7)

(2.8)

for and , where is Euler's constant.

Lemma 2.2.

Let and . Then the following statements are true:

(1)if , then for ;

(2)if , then for ;

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

Proof.

Let and . Then simple computations lead to

(2.9)

(2.10)

(2.11)

(2.12)

(2.13)

(2.14)

(2.15)

• (1) If , then we divide the proof into two cases.

Case 1.

If , then implies that for and for . Thus is strictly increasing on and strictly decreasing on . From (2.10) and we clearly see that there exists such that for and for , which implies that is strictly increasing on and strictly decreasing on . It follows from (2.9) that

(2.16)

for .

Case 2.

If , then we know since . It follows from (2.13) and (2.15) that

(2.17)

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.

If , then from (2.13) and (2.15) we clearly know that for , which implies that is strictly increasing on . Then the properties of and lead to

(2.18)

It follows from (2.12) and (2.18) that there exists such that for and for since as . Hence is strictly decreasing on and strictly increasing on . From (2.10) and we know that there exists such that for and for . Therefore, it follows from (2.9) that

(2.19)

for .

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).

• (3) If and , then from (2.12) we clearly know that . Thus there exists such that for . It follows from (2.10) that , . Since as , we know that there exists such that for , which implies that is strictly increasing on and . Therefore, for and for .

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

Lemma 2.3.

Inequality

(2.20)

holds for .

3. Proof of Theorem 1.1

Proof of Theorem 1.1.

Taking the logarithm of (1.8) and differentiating, then we have

(3.1)

For , it follows from (2.8) that

(3.2)

where

(3.3)

• (1) If , then it follows from (3.1) and (2.20) that

  

  (3.4)

From (3.2) and (3.3) together with Lemma 2.2(1) we clearly see that

(3.5)

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

Conversely, if , then we can divide the set into two subsets: and . Therefore, it follows from Lemma 2.2(2) and (3) that is not strictly logarithmically completely monotonic on for .

• (2) If , then from (3.1) and (2.20) we get

  

  (3.6)

since

(3.7)

For , it follows from (3.2) that

(3.8)

where is defined as (3.3).

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.

Although the upper and lower bounds of Kershaw's inequalities given in (1.11) and (1.12) are not better than those of inequalities in (1.4) and (1.5), the difference between them is close to zero as is large enough. For example,

(3.9)

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.