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 Inequalities and complete monotonicity for the gamma and related functions Commun. Korean Math. Soc. 2019 Vol. 34, No. 4, 1261-1278 https://doi.org/10.4134/CKMS.c180450Published 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 Downloads: Full-text PDF   Full-text HTML