Inequalities involving the gamma and digamma functions

TL;DR

This paper refines upper bounds for inequalities involving the gamma and digamma functions, enhancing previous results and providing tighter estimates for these special functions.

Contribution

It introduces improved upper bounds for inequalities relating gamma and digamma functions, advancing the understanding of their monotonicity and bounds.

Findings

Tighter bounds for gamma function inequalities

Enhanced estimates for digamma function inequalities

Improved monotonicity properties of special functions

Abstract

We improve the upper bounds of the following inequalities proved in [H. Alzer and N. Batir, Monotonicity properties of the gamma function, Appl. Math. Letters, 20(2007), 778-781]. \begin{equation*} exp\left(-\frac{1}{2}\psi\left(x+\frac{1}{3}\right)\right)<\frac{\Gamma(x)}{\sqrt{2\pi}x^xe^{-x}}<exp\left(-\frac{1}{2}\psi(x)\right), \end{equation*} and $$ \frac{1}{2}\psi'(x+1/3))<\log x-\psi(x)<\frac{1}{2}\psi'(x).$$ Here $\Gamma$ is the classical gamma function and $\psi$ is the digamma function.