Inequalities for the inverses of the polygamma functions

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 $\alpha =[(n-1)!{]}^{-1\mathrm{/}n}$ and $\beta =[n!\zeta (n+1){]}^{-1\mathrm{/}n}$, 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.