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arXiv · 1705.06547

Inequalities for the inverses of the polygamma functions

Necdet Batir
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Abstract

We provide an elementary proof of the left side inequality and improve the right inequality in \bigg[\frac{n!}{x-(x^{-1/n}+α)^{-n}}\bigg]^{\frac{1}{n+1}}&<((-1)^{n-1}ψ^{(n)})^{-1}(x) &<\bigg[\frac{n!}{x-(x^{-1/n}+β)^{-n}}\bigg]^{\frac{1}{n+1}}, where $α=[(n-1)!]^{-1/n}$ and $β=[n!ζ(n+1)]^{-1/n}$, which was proved in \cite{6}, and we prove the following inequalities for the inverse of the digamma function $ψ$. \frac{1}{\log(1+e^{-x})}<ψ^{-1}(x)< e^{x}+\frac{1}{2}, \quad x\in\mathbb{R}. The proofs are based on nice applications of the mean value theorem for differentiation and elementary properties of the polygamma functions.

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Necdet Batir. 2017-05-18. Inequalities for the inverses of the polygamma functions. https://arxiv.org/abs/1705.06547

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