  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.
