On a conjecture of a logarithmically completely monotonic function

TL;DR

This paper proves a conjecture related to logarithmically completely monotonic functions on (0,1), extends it to a more general case, and proposes an open problem involving the q-Digamma function.

Contribution

It confirms a specific conjecture on logarithmically completely monotonic functions and generalizes the result, also introducing an open problem involving q-Digamma functions.

Findings

Proved the conjecture for (0,1)

Extended the theorem to a more general setting

Posed an open problem involving q-Digamma functions

Abstract

In this short note we prove a conjecture for the interval $(0,1)$, related to a logarithmically completely monotonic function, presented in \cite{BG}. Then, we extend by proving a more generalized theorem. At the end we pose an open problem on a logarithmically completely monotonic function involving $q$-Digamma function.