Complete monotonicity of a difference between the exponential and trigamma functions

TL;DR

This paper provides a new proof demonstrating that the difference between the exponential function e^{1/t} and the trigamma function is completely monotonic on (0, ∞), using inequalities related to Bessel functions.

Contribution

It introduces a novel proof technique for the complete monotonicity of the difference between e^{1/t} and the trigamma function, based on inequalities involving the modified Bessel function.

Findings

The difference e^{1/t} - ψ'(t) is completely monotonic on (0, ∞).

A new proof method is established using inequalities for the first order modified Bessel function.

The approach provides insights into the monotonicity properties of special functions.

Abstract

In the paper, by directly verifying an inequality which gives a lower bound for the first order modified Bessel function of the first kind, the authors supply a new proof for the complete monotonicity of a difference between the exponential function $e^{1/t}$ and the trigamma function $\psi'(t)$ on $(0,\infty)$.