The Scientific World Journal

Volume 2014 (2014), Article ID 543593, 4 pages

http://dx.doi.org/10.1155/2014/543593

Research Article

Some Inequalities for the -Analogue of the Classical Riemann Zeta Functions and the -Polygamma Functions

Banyat Sroysang

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathumthani 12121, Thailand

Received 30 August 2013; Accepted 12 November 2013; Published 22 January 2014

Academic Editors: A. Barbagallo and Y.-M. Chu

Copyright © 2014 Banyat Sroysang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present the generalizations on some inequalities for the -analogue of the classical Riemann zeta functions and the -polygamma functions.

1. Introduction and Preliminaries

Let and . The -shifted factorial [1–5] is defined by

The -Jackson integral [6] from to is defined by

The -gamma function [6] is defined by

The -digamma [7–9] function is defined by

and may be represented [10] as

where .

The -polygamma function [7] is defined by

and may be represented as

where and .

For any , we denote

where is the integer part of .

The -zeta function [11] is defined by

where . Moreover, in [11], the -analogue of the classical Riemann Zeta function is

where and for any ,

In 2009, Brahim [12] proved the results as follows:

where and . Consider

where .

In 2010, Krasniqi et al. [13] gave an inequality as follows:

where and .

In 2012, Sulaiman [10] gave the inequalities as follows:

where , , , and .

where , , and .

In this paper, we present the generalizations on the above inequalities.

2. Results

Theorem 1. Let , , and be such that and . Then

Proof. By the generalized Hölder inequality,

One can easily check that if we put in Theorem 1 then we get the following.

Corollary 2 (see [10]). Let , , and be such that and . Then

It is easy to notice that if we put and in Theorem 1 then we get the following.

Corollary 3 (see [12]). Let and be such that . Then

Theorem 4. Let , , and . Then

Proof. First, we note that

for all . Then

By Minkowski's inequality,

One can easily check that if we put in Theorem 4 then we get the following.

Corollary 5 (see [10]). Let , , and . Then

Theorem 6. Let be such that . Then

Proof. By the generalized Hölder inequality,

It is easy to notice that if we put in Theorem 6 then we get the following.

Corollary 7 (see [13]). Let be such that . Then

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 referees for their useful comments and suggestions.

References

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