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Inequalities and complete monotonicity for the gamma and related functions

최준상|Chao-Ping Chen
대한수학회 (The Korean Mathematical Society)|2019.10|대한수학회논문집 (Communications of the KMS)|제34권 제4호|pp. 1261-1278|DOI 10.4134/CKMS.c180450

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키워드

gamma functioncompletely monotonic functionpsi (or digamma) functionpolygamma functionslogarithmically completely monotonic functionabsolutely monotonic functionBernstein function
주제분류수학

초록

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.

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.

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