# Inequalities for the inverses of the polygamma functions

## TL;DR

This paper establishes improved inequalities for the inverses of polygamma and digamma functions using elementary calculus techniques, providing tighter bounds and simpler proofs.

## Contribution

It introduces new, sharper inequalities for the inverses of polygamma and digamma functions with elementary proofs, enhancing existing results.

## Key findings

- Derived improved inequalities for polygamma function inverses.
- Established bounds for the inverse of the digamma function.
- Provided elementary proofs using mean value theorem.

## Abstract

We provide an elementary proof of the left side inequality and improve the right inequality in \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}}, where $\alpha=[(n-1)!]^{-1/n}$ and $\beta=[n!\zeta(n+1)]^{-1/n}$, which was proved in \cite{6}, and we prove the following inequalities for the inverse of the digamma function $\psi$. \frac{1}{\log(1+e^{-x})}<\psi^{-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.