Decreasing and complete monotonicity of two functions defined by three derivatives of a completely monotonic function involving the trigamma function

TL;DR

This paper investigates the monotonicity properties of functions involving derivatives of the trigamma function, establishing conditions for complete monotonicity and confirming prior conjectures using Laplace transform techniques.

Contribution

It provides necessary and sufficient conditions for complete monotonicity of functions defined by derivatives of the trigamma function, extending previous results and confirming earlier conjectures.

Findings

Proved decreasing property of a ratio involving derivatives of trigamma function

Established conditions for complete monotonicity of related functions

Generalized known results and confirmed previous conjectures

Abstract

In the paper, by convolution theorem of the Laplace transforms, a monotonicity rule for the ratio of two Laplace transforms, Bernstein's theorem for completely monotonic functions, and other analytic techniques, the authors verify decreasing property of a ratio between three derivatives of a function involving trigamma function and find necessary and sufficient conditions for a function defined by three derivatives of a function involving trigamma function to be completely monotonic. These results confirm previous guesses posed by Qi and generalize corresponding known conclusions.