On a Conjecture of Alzer, Berg, and Koumandos

Mathematics

Ladislav Matejíčka¹ 

[1] Faculty of Industrial Technologies in Púchov, Trenčín University of Alexander Dubček in Trenčín, I. Krasku 491/30, 02001 Púchov, Slovakia;

关键词: completely monotonic functions;    inequality;    polygamma function;    digamma function;   

DOI  :  10.3390/math8061031

来源: DOAJ

【 摘 要 】

In this paper, we find a solution of an open problem posed by Alzer, Berg, and Koumandos: determine $(\alpha ,m)\in {\mathbb{R}}^{+}\times \mathbb{N}$ such that the function $x^{\alpha }\left|{\psi }^{\left(m\right)}\left(x\right)\right|$ is completely monotonic on $(0,\infty ),$ where $\psi \left(x\right)$ denotes the logarithmic derivative of Euler’s gamma function.

【 授权许可】

Unknown