Inequalities of Bernstein type for derivatives of rational functions, and inverse theorems of rational approximation

Let $H_p$ be the Hardy space of functions $f$ that are analytic in the disk $|z|<1$ and let $J^\alpha f$ be the derivative of $f$ of order $\alpha$ in the sense of Weyl. It is shown, for example, that if $r$ is a rational function of degree $n\geqslant1$ with all its poles in the domain $|z|>1$, then $\|J^\alpha r\|_{H_\sigma}\leqslant cn^\alpha\|r\|_{H_p}$, where $p\in(1,\infty]$, $\alpha>0$, $\sigma=(\alpha+p^{-1})^{-1}$ and $c>0$ and depends only on $\alpha$ and $p$.
Bibliography: 32 titles.