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.