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].
Positivity and Fourier integrals over regular hexagon
Some trigonometric integrals involving the log gamma and the digamma function
The $(p,q)$-Analogues of Some Inequalities for the Digamma Function
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