Proofs for a monotonicity result of a function involving the psi and exponential functions

TL;DR

This paper presents two alternative proofs demonstrating that a specific function involving the psi and exponential functions is strictly increasing over the positive real numbers.

Contribution

It provides new mathematical proofs for a monotonicity property of a function involving psi and exponential functions.

Findings

The function is strictly increasing on (0, ∞).

Two different proofs confirm the monotonicity.

The results deepen understanding of special functions' properties.

Abstract

In the note, two alternative proofs are provided for a monotonicity result that the function $\psi(x)+\ln\bigl(e^{1/x}-1\bigr)$ is strictly increasing on $(0,\infty)$, where $\psi(x)$ is the psi function.