Bernstein type inequalities for derivatives of rational functions in $L_p$ spaces for $p<1$

It is shown that if $r$ is a rational function of degree $n$, $0<p<1$, $1/p\notin\mathbb N$, and $r\in L_p(-1,1)$, then for any $s\in\mathbb N$
\begin{equation} \biggl(\int _{-1}^1|r^{(s)}(x)|^\sigma\,dx\biggr)^{1/\sigma} \leqslant cn^s\biggr(\int _{-1}^1|r(x)|^p\,dx\biggr)^{1/p}, \tag{1} \end{equation}
where $\sigma =(s+1/p)^{-1}$ and $c>0$ depends only on $p$ and $s$.
The problem of proving the inequality (1) was posed by Sevast'yanov in 1973. Up to the present, it has been solved for $1<p\leqslant\infty$. In the case $1/p\in\mathbb N$ the inequality is not valid. Some applications of (1) to problems of rational approximation are also given in this paper. Similar questions are considered for the line and circle.