J. Korean Math. Soc. 2021; 58(1): 133-147
Online first article July 22, 2020 Printed January 1, 2021
https://doi.org/10.4134/JKMS.j190874
Copyright © The Korean Mathematical Society.
Khaled Mehrez
University of Tunis El Manar
Our aim in this paper is to derive several new monotonicity properties and functional inequalities of some functions involving the $q$-gamma, $q$-digamma and $q$-polygamma functions. More precisely, some classes of functions involving the $q$-gamma function are proved to be logarithmically completely monotonic and a class of functions involving the $q$-digamma function is showed to be completely monotonic. As applications of these, we offer upper and lower bounds for this special functions and new sharp upper and lower bounds for the $q$-analogue harmonic number harmonic are derived. Moreover, a number of two-sided exponential bounding inequalities are given for the $q$-digamma function and two-sided exponential bounding inequalities are then obtained for the $q$-tetragamma function.
Keywords: Logarithmically completely monotonic function, completely monotonic function, $q$-gamma function, $q$-digamma function, $q$-trigamma function
MSC numbers: 33D05, 33B15, 39B72
2008; 45(2): 559-573
2010; 47(6): 1283-1297
2011; 48(3): 655-667
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