Abstract

In the paper, we discuss a new k-generalization of the Nielsen's β-function. Later, we study the completely monotonicity, convexity and inequalities of the new function.

2010 Mathematics Subject Classification. Primary: 39B12. Secondary: 34A25, 30D05.

1. Introduction

The Nielsen's β-function is defined as ( ^{1, 2, 3, 4, 5})

(1)

(2)

(3)

(4)

where is the digamma or psi function and is the Euler's gamma function. It is well known that

(5)

(6)

The Nielsen's β-function has been deeply researched in past years. Such as, K. Nantomah studied the properties and inequalities of the function in ³, gave some convexities and monotonicity of the function in ⁶, and obtained some convexity, monotonicity and inequalities involving a generalized form of the Wallis's cosine formula in ⁷. The function can be used to calculate some integrals ^{4, 8}. Recently, K. Nantomah studied the properties and inequalities of a p-generalization of the Nielsen's function in ⁵. In this paper, we investigate a k-generalization of the Nielsen's β-function function. The notations and

The k-analogue of the gamma function is defined as

(7)

where and .

Remark 1.1. For and the k-analogue of the Gamma function satisfies

1.

2.

2. The New Generalized Nielsen's β-function

In this part, we define a k-generalization of the Nielsen's β-function and study some of its properties.

Definition 2.1. We define the k-generalization of the Nielsen's β-function as

(8)

(9)

(10)

(11)

where is the digamma or psi function and is defined in (7).

Some special values of the k-generalization function

can be calculated, for example

Theorem 2.2. The k-generalization function satisfies the equality for and

(12)

Proof.

The proof is finished.

By (12), we achieve

(13)

and the relationship

(14)

Theorem 2.3. For and

(1) is positive and decreasing if is even;

(2) is negative and increasing if is odd;

(3) is decreasing for any and

Proof. Using (9), we get the recursive formula

(15)

Then we can achieve the conclusion.

Corollary 2.4. The k-generalization function has the following properties for and

(1) is completely monotonic;

(2) is completely monotonic if is even;

(3) is completely monotonic if is odd.

Proof. Using (15), we get

(16)

Then we achieve the conclusion.

Theorem 2.5. The k-generalization function is logarithmical convex for and

Proof. Let and for and , by the Hölder's inequality, we can get

From the definition of logarithmical convex, we can achieve the conclusion.

Based on the above, we could infer the followings:

Corollary 2.6. The k-generalization function satisfies

(17)

for , and .

Proof. Using (15) and the similar procedure in Theorem 2.5, we can get the inequality (17).

Remark 2.7. When is even in Corollary 2.6, then satisfies

Especially, let a = b = 2, x = y and n = m + 2 in Corollary 2.6, we achieve the Turan-Type inequality

(18)

Moreover, if m = 0 in (18), we have the famous inequality

(19)

which implies the function is increasing for

Corollary 2.8. The k-generalization function satisfies

(20)

(21)

for

Proof. Since is decreasing, we get

Let in Theorem 2.5, we achieve

then we can get

By the basic AM-GM inequality, we get

These imply that the inequalities (20) and (21) are right.

Corollary 2.9. The formula

(22)

satisfies for and .

Proof. Since is decreasing and positive, so which implies

Let and in (20), we achieve

then we can get

These imply that the inequality (22) is right.

Theorem 2.10. For and satisfies the inequality

(23)

Proof. Theorem 2.5 implies that is convex. So satisfies the inequality

(24)

In view of (11) and setting (24) becomes

(25)

Taking the exponent of the inequality, we achieve the desired result.

Theorem 2.11. For and the function is completely monotonic, decreasing and convex.

Proof. By calculating, we can get

(26)

By the convolution theorem for Laplace transforms, we achieve

where

(27)

(28)

and Then, for , we achieve

So

(29)

which implies that is completely monotonic.

Letting in (29), we achieve

That is

(30)

which implies is decreasing, and let in (29), we can get

(31)

which implies that is convex.

Corollary 2.12. The function is increasing and convex for .

Proof. Based on (29), (31) and Theorem 2.3, we can achieve

(32)

and

(33)

which implies that is increasing and convex for .

Corollary 2.13. For and the inequality

(34)

is right.

Proof. For , set

then

Since and is increasing, we can get , which implies is increasing. So, for we can achieve

hence

Theorem 2.14. For and the function

(35)

Hence,

(1) and

(2) is decreasing.

Proof. By Theorem 2.3, we could infer

(36)

and

(37)

so, for and

(38)

where . Then

(39)

and

where

By the convolution theorem for Laplace transforms, we achieve

where

(40)

(41)

and . Then, for we achieve

(42)

which implies that is decreasing.

3. Conclusion

In the paper, we defined a new k-generalization function of the Nielsen's β-function, proved the new k-generalization function is convex, decreasing and completely monotonic as well as conducted some inequalities related with the function. The results can be used to evaluate or estimate some integrals. Moreover, the conclusions would play important role in the further study of the function.

Thanks

The authors are grateful to anonymous referees and the editor for their careful corrections and valuable comments on the original version of this paper.

Funding

This work was supported by National Natural Science Foundation of China (Grant No.11601036, Grant No.11701320 and 11705122), the Science and Technology Foundations of Shandong Province (Grant No. J16li52, J14li54 and J17KA161) and Science Foundations of Binzhou University (Grant No. BZXYFB20150903 and BZXYL1704).

References