Inequalities and complete monotonicity for the gamma and related functions
Commun. Korean Math. Soc. 2019 Vol. 34, No. 4, 1261-1278
Published online October 31, 2019
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
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