Some inequalities for the trigamma function in terms of the digamma function

TL;DR

This paper derives new inequalities relating the trigamma function to the digamma function using exponential functions, extending existing results with methods based on asymptotic theory.

Contribution

It introduces three new double inequalities for the trigamma function in terms of the digamma function, expanding the set of known bounds.

Findings

Established three new inequalities for the trigamma function.

Extended known results on inequalities involving special functions.

Used asymptotic theory to derive bounds for large arguments.

Abstract

In the paper, the authors establish three kinds of double inequalities for the trigamma function in terms of the exponential function to powers of the digamma function. These newly established inequalities extend some known results. The method in the paper utilizes some facts from the asymptotic theory and is a natural way to solve problems for approximating some quantities for large values of the variable.