Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2019; 34(4): 1261-1278

Online first article August 27, 2019      Printed October 31, 2019

https://doi.org/10.4134/CKMS.c180450

Copyright © The Korean Mathematical Society.

Inequalities and complete monotonicity for the gamma and related functions

Chao-Ping Chen, Junesang Choi

Henan Polytechnic University; Dongguk University

Abstract

It is well-known that if $\phi ''>0$ for all $x$, $\phi(0)=0$, and $\phi /x$ is interpreted as $\phi '(0)$ for $x=0$, then $\phi /x$ increases for all $x$. This has been extended in [Complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions, J. Math. Anal. Appl. \textbf{336} (2007), 812--822]. In this paper, we extend the above result to the very general cases, and then use it to prove some (logarithmically) completely monotonic functions related to the gamma function. We also establish some inequalities for the gamma function and generalize some known results.

Keywords: gamma function, psi (or digamma) function, polygamma functions, completely monotonic function, logarithmically completely monotonic function, absolutely monotonic function, Bernstein function

MSC numbers: Primary 33B15, 26A48