Monotonic and Logarithmically Convex Properties of a Function Involving Gamma Functions

Abstract

Using the series-expansion of digamma functions and other techniques, some monotonicity and logarithmical concavity involving the ratio of gamma function are obtained, which is to give a partially affirmative answer to an open problem posed by B.-N.Guo and F.Qi. Several inequalities for the geometric means of natural numbers are established.

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].

In recent years, many monotonicity results and inequalities involving the Gamma and incomplete Gamma functions have been established. This article is stimulated by an open problem posed by Guo and Qi in [2]. The extensions and generalizations of this problem can be found in [3–5] and some references therein.

Using Stirling formula, for all nonnegative integers , natural numbers and , an upper bound of the quotient of two geometrical means of natural numbers was established in [4] as follows:

(1.2)

and the following lower bound was appeared in [2, 5]:

(1.3)

Since as a generalization of inequality (1.3), the following monotonicity result was obtained by Guo and Qi in [2]. The function

(1.4)

is decreasing with respect to on for fixed Hence, for positive real numbers and , we have

(1.5)

Recently, in [6], Qi and Sun proved that the function

(1.6)

is strictly increasing with respect to for all

Now, we generalize the function in (1.4) as follows: for positive real numbers and , , let

(1.7)

The aim of this paper is to discuss the monotonicity and logarithmical convexity of the function with respect to parameter .

For convenience of the readers, we recall the definitions and basic knowledge of convex function and logarithmically convex function.

Definition 1.1.

Let be a convex set, is called a convex function on if

(1.8)

for all , and is called concave if is convex.

Definition 1.2.

Let be a convex set, is called a logarithmically convex function on if is convex on , and is called logarithmically concave if is concave.

The following criterion for convexity of function was established by Fichtenholz in [7].

Proposition 1.3.

Let be a convex set, if have continuous second partial derivatives, then is a convex (or concave) function on if and only if is a positive (or negative) semidefinite matrix for all , where

(1.9)

and for , .

Notation.

In Definitions 1.1, 1.2 and Proposition 1.3, we denote by the points (or vectors) of , and denote by the real variables in the later.

Our main results are Theorems 1.4 and 1.5.

Theorem 1.4.

• (1) For any fixed , is strictly increasing (or decreasing, resp.) with respect to on if and only if (or , resp.);

• (2) For any fixed , is strictly increasing with respect to on if and only if .

Theorem 1.5.

• (1) If , then is logarithmically concave with respect to ;

• (2) If is a convex set with nonempty interior and , then is neither logarithmically convex nor logarithmically concave with respect to on .

The following two corollaries can be derived from Theorems 1.4 and 1.5 immediately.

Corollary 1.6.

If , then

(1.10)

Remark 1.7.

Inequality (1.3) can be derived from Corollary 1.6 if we take . Although we cannot get the inequality (1.2) exactly from Corollary 1.6, but we can get the following inequality which is close to inequality (1.2):

(1.11)

Corollary 1.8.

If , then

(1.12)

Remark 1.9.

We conjecture that the inequality (1.2) can be improved if we can choose two pairs of integers and properly.

2. Lemmas

It is well known that the Bernoulli numbers is defined [8] in general by

(2.1)

In particular, we have

(2.2)

In [9], the following summation formula is given:

(2.3)

for nonnegative integer , where denotes the Euler number, which implies

(2.4)

Recently, the Bernoulli and Euler numbers and polynomials are generalized in [10–13]. The following two Lemmas were established by Qi and Guo in [3, 14].

Lemma 2.1 (see [3]).

For real number and natural number , one has

(2.5)

(2.6)

(2.7)

(2.8)

Lemma 2.2 (see [14]).

Inequalities

(2.9)

,

(2.10)

hold in for .

Lemma 2.3.

Let , then the following statements are true:

(1) if , then for ;

(2) if , then for .

Proof.

• (1) Making use of (2.6) we get

  

  (2.11)

for any fixed .

Since and , we have

(2.12)

for all .

Therefore, Lemma 2.3(1) follows from (2.11) and (2.12).

• (2) If , then (2.12) leads to

  

  (2.13)

for .

Therefore, Lemma 2.3(2) follows from (2.11) and (2.13).

Lemma 2.4.

If , then for .

Proof.

It is easy to see that

(2.14)

for all .

Let , then

(2.15)

(2.16)

(2.17)

for . On the other hand, from (2.10) we know that is strictly decreasing on .

Therefore, Lemma 2.4 follows from (2.14)–(2.17).

Remark 2.5.

Let

(2.18)

Then simple computation shows that

(2.19)

Lemma 2.6.

Let then the following statements are true:

(1) if , then

(2.20)

for ;

(2) if then

(2.21)

for .

Proof.

Let

(2.22)

Then it is not difficult to verify

(2.23)

(2.24)

(2.25)

• (1) If , then making use of Lemmas 2.2, 2.4 and (2.25) we get

  

  (2.26)

for .

Let , and . Then simple computation leads to

(2.27)

(2.28)

(2.29)

(2.30)

for all .

It is well known that , where is the Euler's constant. From this we get

(2.31)

From Lemma 2.2, (2.27)–(2.29), (2.31) and the assumption , we conclude that

(2.32)

Therefore, Lemma 2.6(1) follows from (2.23)–(2.26), (2.30), and (2.32).

• (2) If , then making use of (2.8), Lemma 2.4 and (2.25) we obtain

  

  (2.33)

Let

(2.34)

Then

(2.35)

for by Lemma 2.2, and

(2.36)

for .

Therefore, Lemma 2.6(2) follows from (2.23)–(2.25) and (2.33)–(2.36).

3. Proofs of Theorems 1.4 and 1.5

Proof of Theorem 1.4.

• (1) Let and , then

  

  (3.1)

The following three cases will complete the proof of Theorem 1.4(1).

Case 1.

If , then (3.1) and Lemma 2.2 imply

(3.2)

for .

From (3.2) and the fact that for all we know that is strictly increasing with respect to on for any fixed .

Case 2.

If , then (3.1) and (2.7) imply

(3.3)

for , where and .

From (3.3) and the fact that for all we know that is strictly decreasing with respect to on for any fixed .

Case 3.

If , let

(3.4)

Then

(3.5)

(3.6)

for .

It is obvious that (3.6) implies

(3.7)

The continuity of with respect to for any fixed and (3.7) imply that there exists such that

(3.8)

for .

From (3.5), (3.8) and we know that is strictly decreasing with respect to on for .

On the other hand, making use of (2.5) and (2.6) we have

(3.9)

where

(3.10)

for and .

Equation (3.9) implies that there exists such that

(3.11)

for .

Hence, from (3.11) we know that is strictly increasing with respect to on for .

• (2) Since

  

  (3.12)

then, Theorem 1.4(2) follows from (3.12) and Lemma 2.3.

Proof of Theorem 1.5.

Let and , then simple calculation yields

(3.13)

(3.14)

(3.15)

where , and are defined in Remark 2.5 and Lemma 2.6.

According to the Definition 1.2 and Proposition 1.3, to prove Theorem 1.5 we need only to show that

(3.16)

(3.17)

for and , and

(3.18)

for and .

Next, let then

(3.19)

(3.20)

for by Lemma 2.2 and .

Therefore, (3.16) follows from (3.19) and (3.20), and (3.17) and (3.18) follow from Lemma 2.6. The proof of Theorem 1.5 is completed.