On functional inequalities for the psi function

TL;DR

This paper investigates the monotonicity and inequalities related to the psi function, which is the logarithmic derivative of the gamma function, providing new insights into its properties.

Contribution

It introduces new monotonicity results and inequalities involving the psi function, expanding understanding of its behavior and applications.

Findings

Established monotonicity of specific psi function ratios

Derived several new inequalities involving the psi function

Enhanced theoretical understanding of the psi function's properties

Abstract

In this note we study the monotonicity of the function $x\mapsto \psi(1 +bx)^a/\psi(1 + ax)^b$. We also give the several inequalities involving the psi function, whic is the logarithmic derivative of the gamma function.