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.