Complete monotonicity of some functions involving polygamma functions

Details Complete monotonicity of some functions involving polygamma functions Complete monotonicity of some functions involving polygamma functions

2009-05-17

arxiv.org/abs/0905.2732v1

Feng Qi, Senlin Guo and Bai-Ni Guo, Complete monotonicity of some functions involving polygamma functions, Journal of Computat

Mathematics

Classical Analysis and ODEs

14 pages

Scientific paper

10.1016/j.cam.2009.09.044

In the present paper, we establish necessary and sufficient conditions for the functions $x^\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert$ and $\alpha\bigl\lvert\psi^{(i)}(x+\beta)\bigr\lvert-x\bigl\lvert\psi^{(i+1)}(x+\beta)\bigr\lvert$ respectively to be monotonic and completely monotonic on $(0,\infty)$, where $i\in\mathbb{N}$, $\alpha>0$ and $\beta\ge0$ are scalars, and $\psi^{(i)}(x)$ are polygamma functions.

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