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.
Chao-Ping Chen, Junesang Choi
Henan Polytechnic University; Dongguk University
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
2018; 33(2): 549-560
2018; 33(1): 143-155
2016; 31(3): 591-601
2015; 30(4): 439-446
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