Sharp inequalities for the psi function and harmonic numbers

TL;DR

This paper establishes precise inequalities for the psi function and harmonic numbers, improving previous bounds and providing new insights into their behavior.

Contribution

It introduces new sharp inequalities for the psi function and harmonic numbers, enhancing existing approximation results and offering refined bounds.

Findings

New sharp bounds for the psi function

Improved inequalities for harmonic numbers

Enhanced approximation accuracy

Abstract

In this paper, two sharp inequalities for bounding the psi function $\psi$ and the harmonic numbers $H_n$ are established respectively, some results in [I. Muqattash and M. Yahdi, \textit{Infinite family of approximations of the Digamma function}, Math. Comput. Modelling \textbf{43} (2006), 1329\nobreakdash--1336.] are improved, and some remarks are given.