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

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].