A class of completely monotonic functions involving divided differences of the psi and polygamma functions and some applications

TL;DR

This paper introduces a class of completely monotonic functions based on divided differences of psi and polygamma functions, deriving various inequalities and properties related to gamma functions, factorial ratios, and volume ratios.

Contribution

It establishes new completely monotonic functions involving divided differences of psi and polygamma functions and applies them to derive multiple inequalities and bounds in special functions and geometric contexts.

Findings

Proved a class of functions involving divided differences are completely monotonic.

Derived monotonicity and convexity properties of gamma-related functions.

Recovered and generalized several inequalities involving factorial ratios, error functions, and volume ratios.

Abstract

A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving ratio of two gamma functions and originating from establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.