Sharp inequalities for polygamma functions

Abstract

The main aim of this paper is to prove that the double inequality \frac{(k-1)!}{\Bigl\{x+\Bigl[\frac{(k-1)!}{|ψ^{(k)}(1)|}\Bigr]^{1/k}\Bigr\}^k} +\frac{k!}{x^{k+1}}<\bigl|ψ^{(k)}(x)\bigr|<\frac{(k-1)!}{\bigl(x+\frac12\bigr)^k}+\frac{k!}{x^{k+1}} holds for $x>0$ and $k\in \mathbb{N}$ and that the constants $[\frac{(k-1)!}{|{\psi }^{(k)}(1)|}{]}^{1/k}$ and $\frac{1}{2}$ are the best possible. In passing, some related inequalities and (logarithmically) complete monotonicity results concerning the gamma, psi and polygamma functions are surveyed.

11 pages

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