Inequalities for the inverses of the polygamma functions

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Abstract

We provide an elementary proof of the left-hand side of the following inequality and give a new upper bound for it.
$$\begin{aligned} \bigg [\frac{n!}{x-(x^{-1/n}+\alpha )^{-n}}\bigg ]^{\frac{1}{n+1}}&<((-1)^{n-1}\psi ^{(n)})^{-1}(x) \\&<\bigg [\frac{n!}{x-(x^{-1/n}+\beta )^{-n}}\bigg ]^{\frac{1}{n+1}}, \end{aligned}$$
where \(\alpha =[(n-1)!]^{-1/n}\) and \(\beta =[n!\zeta (n+1)]^{-1/n}\), which was proved in Batir (J Math Anal Appl 328:452–465, 2007), and we prove the following inequalities for the inverse of the digamma function \(\psi \).
$$\begin{aligned} \frac{1}{\log (1+e^{-x})}<\psi ^{-1}(x)< e^{x}+\frac{1}{2}, \quad x\in \mathbb {R}. \end{aligned}$$
The proofs are based on nice applications of the mean value theorem for differentiation and elementary properties of the polygamma functions.

Keywords

Inverse of digamma function Mean value theorem Gamma function Polygamma functions Inequalities 

Mathematics Subject Classification

Primary 33B15 Secondary 26D07 

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Notes

Acknowledgements

I would like to thank the referee for his/her thorough review and highly appreciated the comments and suggestions.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Sciences and ArtsNevşehır Hacı Bektaş Veli UniversityNevşehırTurkey

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