On the positivity of a certain function related with the Digamma function

TL;DR

This paper proves the positivity of a specific function involving derivatives related to the Digamma function for all large x and natural n, improving previous results in mathematical analysis.

Contribution

It establishes a new positivity result for a function involving derivatives of a ratio related to the Digamma function, extending prior work.

Findings

The function rac{x^n}{1-e^{-x}}erivative is positive for all x > e2b9 and n b1 natural numbers.

The result improves previous bounds and extends the understanding of related special functions.

The proof covers all x in (e2b9, b6b0) and all n in b1 b7b7

Abstract

It is proved that $$\left(\frac{x^n}{1-e^{-x}}\right)^{(n)}>0$$ for all $x\in (\log 2, \infty)$ and $n\in \mathbb{N}$, which improves the result of [Al-Musallam and Bustoz in Ramanujan J. 11 (2006) 399-402].