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 Inequalities and complete monotonicity for the gamma and related functions Commun. Korean Math. Soc.Published online July 1, 2019 Chao-ping Chen and 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]. Here, in this paper, we extend the above results to the very general cases and establish Theorem \ref{09Jlem1}. In order to show how useful our main result Theorem \ref{09Jlem1} is, we apply 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 function- s; Completely monotonic function; Logarithmically completely monotonic function; Absolutely monotonic function; Bernstein function MSC numbers : 33B15; 26A48 Full-Text :