#### Abstract

We prove that the function is strictly logarithmically completely monotonic on if and is strictly logarithmically completely monotonic on if , where and

#### 1. Introduction

It is well known that the classical Euler’s gamma function is defined for as The logarithmic derivative of defined by is called the psi or digamma function and for are known as the polygamma or multigamma functions. 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.

For extension of these functions to complex variable and for basic properties, see [1]. Over the past half century, many authors have established inequalities and monotonicity for these functions (see [2–22]).

Recall that a real-valued function is said to be completely monotonic on if has derivatives of all orders on and for all and . Moreover, is said to be strictly completely monotonic if inequality (1.3) 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 for all and . Moreover, is said to be strictly logarithmically completely monotonic if inequality (1.4) is strict.

Recently, the completely monotonic or logarithmically completely monotonic functions have been the subject of intensive research. There has been a lot of literature about the (logarithmically) completely monotonic functions related to the gamma function, psi function, and polygamma function, for example, [17, 18, 23–37] and the references therein. In 1997, Merkle [38] proved that is strictly log-concave on . Later, Chen [39] showed that is strictly logarithmically completely monotonic on . In [40], Li and Chen proved that is strictly logarithmically completely monotonic on for , and is strictly logarithmically completely monotonic on for . Qi et al. in their article [41] showed that is strictly logarithmically complete monotonic on for , and is strictly logarithmically complete monotonic on for .

The aim of this paper is to discuss the logarithmically complete monotonicity properties of the functions and on where and . The function is the deformation of the functions in [40, 41] with respect to the parameters and . We show that the properties of logarithmically complete monotonic are also true for suitable extensions of near by two lines and , which generalizes the results of [40, 41].

For , we define two binary functions as follows:

For convenience, we need to define five subsets of and refer to Figure 2,

We summarize the result as follows.

Theorem 1.1. *Let , , and be defined as (1.5); then the following statements are true: *(1)* is strictly logarithmically completely monotonic on if *(2)* is strictly logarithmically completely monotonic on if **Note that is the constant 1 for since .*

#### 2. Lemmas

In order to prove our Theorem 1.1, we need two lemmas which we present in this section.

We consider and defined as (1.6) and discuss the properties for these functions, see Figure 1 more clearly.

##### 2.1. The Properties of Function

The function can be interpreted as a quadric equation with respect to . Let where , , and its discriminant function

If , then it is easy to see that for .

Let , be two real roots of with ; then we claim that . Indeed, From (2.5)–(2.7), we know that has only one root , which is Moreover, for and for , which implies that is strictly decreasing on and strictly increasing on . An easy computation shows that , , and . Combining with (2.4), there exist two real roots such that . Furthermore, we conclude that for or and for .

If , then since and .

If , then , , which implies .

If or , then . We can solve two roots of the equation , which are For , we know that for and for or . For , we know that for and for . Moreover, we see that as and as .

##### 2.2. The Properties of Function

The function can also be interpreted as a quadric equation with respect to . Let where , , and its discriminant function

If , then we have for .

If , then a simple calculation leads to for . This implies that . Notice that , , and ; for , then we have .

If , then we can solve the roots of the equation but only one of the roots is positive, that is,

Therefore, we conclude that for and for . Moreover, it is easy to see that as and as .

Finally, we calculate an intersection point of and , that is, the point

Lemma 2.1. *The psi or digamma function, the logarithmic derivative of the gamma function, and the polygamma functions can be expressed as
**
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 , or , , then there exist such that for and for ; *(4)*if , , then there exist such that for and for .*

*Proof. *Let , and . Then simple calculations lead to

(1) If then we divide the proof into two cases. Note that , see Figure 2.*Case 1. *If then , , and it follows from (2.21) that

Therefore, for follows from (2.17), (2.18) together with (2.27).*Case 2. *If , then , , and . It follows from that and then (2.20) and (2.22) together with (2.24) lead to
This could not happen together for all qualities of (2.28)–(2.31) since the qualities of (2.29) and (2.30) hold only for , while the qualities of (2.29) and (2.30) hold only for .

Therefore, for follows from (2.17) and (2.18) together with (2.28)–(2.31).

(2) If then we divide the proof into three cases.*Case 1. *If then and . From (2.26), we clearly see that

In terms of the properties of , we know that for lying on the left-side of the green curve, see Figure 1. From (2.24), we see that
Combining (2.32) with (2.33) we get that is strictly increasing on .

If , then and follow from (2.22), which implies that is strictly increasing in . Thus we can obtain

If , then it follows from or that there exists such that for and for . Hence, is strictly decreasing in and strictly increasing in . Then we can obtain
Finally, we conclude that for follows from (2.17), (2.18) together with (2.34), (2.35).*Case 2. *If then and . It follows from (2.21) that

Therefore, for follows from (2.17), (2.18) together with (2.36).*Case 3. *If then and . From (2.26), we know that
In terms of the location of we know that . From (2.24), we see that
It follows from (2.37) and (2.38) that is strictly increasing on .

If , then and follow that from (2.22), which implies that is strictly increasing on . From (2.20) and (2.21), we see that
Thus there exists such that for and for , which implies that is strictly decreasing on and strictly increasing on . It follows from (2.18) and that such that for and for , which implies that is strictly decreasing on and strictly increasing on . Therefore, it follows from (2.17) and that
for .

If , then there exists such that for and for follows from or . This leads to being strictly decreasing in and strictly increasing in . From (2.20), we clearly see that

For special case of , that is, and , it follows from (2.41) and (2.21) that
which implies that for follows from (2.17) and (2.18).

For , it follows from (2.38) and that there exists such that for and for . Making use of the same arguments as the case of , then for follows from (2.17).

(3) If , or , , then we have

From (2.20), we know that

It follows from (2.44) that there exists such that for , which implies that is strictly increasing on . Therefore, for follows from (2.17) and (2.18).

From (2.43), we know that there exists such that for .

(4) If , , then we have

From (2.15), we know that

Making use of (2.45) and (2.46) together with the same arguments as in Lemma 2.2(3), we know that there exist such that for and for .

#### 3. Proof of Theorem 1.1

*Proof of Theorem 1.1. *From (2.15), we have
where

(1) If then from (3.1) and (3.2) together with Lemma 2.2(1) we clearly see that
Therefore, is strictly logarithmically completely monotonic on following from (3.3).

(2) If then from (3.1) we can get
where is defined as (3.2).

Therefore, is strictly logarithmically completely monotonic on following from (3.4) and Lemma 2.2 (2).

*Remark 3.1. *Note that neither nor is strictly logarithmically completely monotonic on for following from Lemma 2.2 (3) and (4), it is known that the logarithmically completely monotonicity properties of and are not completely continuously depended on and .

*Remark 3.2. *Compared with Theorem 9 of [40], we can also extend onto one component of its boundaries, which is
Then is strictly logarithmically completely monotonic on for

#### Acknowledgment

The first author is supported by the China-funded Postgraduates Studying Aboard Program for Building Top University.