Inequalities for the inverses of the polygamma functions

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.