A Class of Logarithmically Completely Monotonic Functions and Their Applications

Abstract

We study the recent investigations on a class of functions which are logarithmically completely monotonic. Two open problems are also presented.

1. Introduction

Recall [1] that a positive function is said to be logarithmically completely monotonic (LCM) on an open interval if has derivatives of all orders on and for all ,

LCM functions are related to completely monotonic (CM) functions [2], strongly logarithmically completely monotonic (SLCM) functions [3], almost strongly completely monotonic (ASCM) functions [3], almost completely monotonic (ACM) functions [4], Laplace transforms, and Stieltjes transforms and have wide applications. It is evident that the set of SLCM functions is a nontrivial subset of the set of LCM functions, which is a nontrivial subset of the set of CM functions, and that the set of CM functions is a nontrivial subset of the set of ACM functions. It was established [3] that the set of SLCM functions is a nontrivial subset of the set of ASCM functions and that the set of SLCM functions on the interval is disjoint with the set of strongly completely monotonic (SCM) functions (see [5] for its definition) on the interval .

It is well known that the classical Euler gamma function is defined for by The logarithmic derivative of , denoted by is called psi function, and for are called polygamma functions.

For and , define which is encountered in probability and statistics.

Since    is logarithmically completely monotonic if and only if is logarithmically completely monotonic and    is logarithmically completely monotonic if and only if is logarithmically completely monotonic, we only need to study the logarithmically complete monotonicity of the function

In [6, Theorem 3.2], it was proved that the function is decreasing and logarithmically convex from onto and that the function is increasing and logarithmically concave from onto .

In [7, Theorem 1], for showing for monotonic properties of the functions and on the interval were obtained.

In [8, Theorem 2], it was presented that the function is decreasing on the interval for if and only if and increasing on the interval if and only if

In [9], after proving the logarithmically completely monotonic property of the functions and , in virtue of Jensen’s inequality for convex functions, the upper and lower bounds for the Gurland’s ratio were established: for positive numbers and , the inequality holds true, where the middle term in (10) is called Gurland’s ratio [10].

In [11] the authors proved the following result.

Theorem 1 (see [11]). If then the function is logarithmically completely monotonic on the interval .

The necessary and sufficient conditions for the functions and to be logarithmically completely monotonic on the interval were also given in [11].

Using monotonic properties of the functions and , the inequality (6) was extended (see [11, Remark 1]) from to

In [12] the authors proved the following results.

Theorem 2 (see [12]). If and , then the function is logarithmically completely monotonic on the interval .

Theorem 3 (see [12]). For , a necessary condition for the function to be logarithmically completely monotonic on the interval is that

Theorem 4 (see [12]). For , a necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval is that

As direct consequences of the above results, the following Kečkić-Vasić-type inequality is deduced.

Theorem 5 (see [12]). Let and be positive numbers with .For , the following inequality  holds true if and only if , where  is the identric or exponential mean.For , the inequality (16) holds true also if .

In [13], the following result was established.

Theorem 6 (see [13]). For , if then the function is logarithmically completely monotonic on the interval .
For , if then the function is logarithmically completely monotonic on the interval .

From Theorem 6 we can directly obtain the following new result.

Corollary 7. For , if then the function is logarithmically completely monotonic on the interval .
For , if then the function is logarithmically completely monotonic on the interval .
For , if then the function is logarithmically completely monotonic on the interval .

A necessary and sufficient condition is obtained in [13] as follows.

Theorem 8 (see [13]). For a necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval is that

Regarding the logarithmically complete monotonicity for the function and their applications. In [14], the authors proved the following results.

Theorem 9 (see [14]). If the function is logarithmically completely monotonic on the interval , then either or

Theorem 10 (see [14]). For the necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval is that

As first application, the following inequalities are derived by using logarithmically completely monotonic properties of the function on the interval .

Theorem 11 (see [14]). For , double inequalities hold true on the interval .
When , inequalities hold true on the interval for .
When , inequalities hold true on the interval for .
When , inequalities hold true on the interval for .

As second application, the following inequalities are derived by using logarithmically convex properties of the function on .

Theorem 12 (see [14]). Let and Suppose also that If either or then If then the inequality (37) reverses.

As final application, the following inequality can be derived by using the decreasingly monotonic property of the function on .

Theorem 13 (see [14]). If then holds true for with , where , defined by (17), is the identric or exponential mean.

The following results were shown in [15].

Theorem 14 (see [15]). For a sufficient condition for the function to be logarithmically completely monotonic on the interval is that

Remark 15. From Theorems 9 and 14 we see that the necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval is that This result is Theorem 2 in [11]. Here we recovered it.

Theorem 16 (see [15]). Let Then the necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval is that

The following results are applications of the above theorems.

Theorem 17 (see [15]). When the following inequalities hold true on the interval .

Theorem 18 (see [15]). Let and Suppose also that If then

Theorem 19 (see [15]). If then where in (53) , defined by (17), is the identric or exponential mean.

2. Open Problems

2.1. Open Problem 1

From Theorem 8 we have already known, for a necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval .

For what is a necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval

Already Known. Theorem 3 gave a necessary condition; Theorem 6 provided a sufficient condition.

2.2. Open Problem 2

From Remark 15, Theorems 10 and 16 we have already known, for a necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval .

For what is a necessary and sufficient condition for the function to be logarithmically completely monotonic on the interval

Already Known. Theorem 9 gave a necessary condition; Theorem 14 provided a sufficient condition.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author would like to thank the editor and the reviewers for their valuable comments and suggestions to improve the quality of this paper. The present investigation was supported, in part, by the Natural Science Foundation of China under Grant no. 11326167.