期刊
AEQUATIONES MATHEMATICAE
卷 95, 期 3, 页码 569-588出版社
SPRINGER BASEL AG
DOI: 10.1007/s00010-020-00767-6
关键词
Gamma function; q-Gamma function; Digamma function; Completely monotonic function; Bernstein function; Logarithmic complete monotonicity; Sherman's theorem
This paper investigates the conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions, provides necessary and sufficient conditions, and gives new examples of logarithmically completely monotonic gamma ratios. The results are then applied to study monotonicity of some gamma ratios and rational functions.
We investigate conditions for logarithmic complete monotonicity of product ratios of gamma and q-gamma functions whose arguments are linear functions of the variable. We give necessary and sufficient conditions in terms of nonnegativity of a certain explicitly written measure in the q case and of a certain elementary function in the classical q=1 case. In the latter case we further provide simple new sufficient conditions leading to many new examples of logarithmically completely monotonic gamma ratios. Finally, we apply some of our results to study monotonicity of some gamma ratios and rational functions.
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