Sharp inequalities for polygamma functions

TL;DR

This paper establishes sharp bounds for polygamma functions, proving the best possible constants in inequalities, and surveys related inequalities and monotonicity properties of gamma and polygamma functions.

Contribution

The paper provides the first proof of optimal bounds for polygamma functions with precise constants and surveys related monotonicity and inequality results.

Findings

Established sharp inequalities for polygamma functions with best constants.

Proved the inequalities hold for all positive x and natural k.

Surveyed related inequalities and monotonicity properties of gamma and 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)!}{|\psi^{(k)}(1)|}\Bigr]^{1/k}\Bigr\}^k} +\frac{k!}{x^{k+1}}<\bigl|\psi^{(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 $\Bigl[\frac{(k-1)!}{|\psi^{(k)}(1)|}\Bigr]^{1/k}$ and $\frac12$ are the best possible. In passing, some related inequalities and (logarithmically) complete monotonicity results concerning the gamma, psi and polygamma functions are surveyed.