Bernstein-type inequalities for the derivatives of rational functions in $L_{p}$-spaces, $0<p<1$, on Lavrent′ev curves

Abstract:

Let $S$ be a simple or a closed Lavrent′ev curve on the complex plane, let $0<p<1$ with $1/p \not \in \mathbb {N}$, and let $s\in \mathbb {N}$. It is shown that for an arbitrary rational function $r$ of degree $n$ such that $|r|^{p}$ is integrable on $S$ the following inequality is fulfilled: \begin{equation*} \bigg (\int _{S}|r^{(s)}(z)|^{\sigma } |dz|\bigg )^{1/\sigma } \le cn^{s} \bigg (\int _{S} |r(z)|^{p} |dz|\bigg )^{1/p} , \end{equation*} where $1/\sigma =s+1/p$, and $c>0$ depends only on $S, p$, and $s$.

Earlier (in 1995) this result was obtained by the author and Stahl for the segment and the circle. The inequality is used to deduce an inverse rational approximation theorem in the Smirnov class $E_{p}$. Other rational approximation problems in $L_{p}$ and $E_{p}$ are also treated.