Complete monotonicity of some functions involving polygamma functions

Abstract

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

14 pages

References in corpus (2)

• Some logarithmically completely monotonic functions related to the gamma function
• Some Monotonicity Properties of Gamma and $q$-gamma Functions

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