Conjecture concerning a completely monotonic function

TL;DR

This paper presents a conjecture about the complete monotonicity of a specific function class, based on numerical evidence, and identifies the critical parameter value where the property holds.

Contribution

It introduces a new conjecture on the complete monotonicity of functions involving exponential and power expressions, supported by numerical computations.

Findings

Numerical evidence suggests a critical value of parameter a.

The critical value for complete monotonicity is approximately determined.

The exact symbolic form of the critical value remains unknown.

Abstract

Based on a sequence of numerical computations, a conjecture is presented regarding the class of functions $H(x;a)=\exp(a)-(1+a/x)^x$, and the open problem of determining the values of $a$ for which the functions are completely monotonic with respect to $x$. The critical value of $a$ is determined here to sufficient accuracy to show that it is not a simple symbolic quantity.