An integral estimate for the derivative of a rational function

Let there be given numbers $\alpha,q,\lambda,p$ and $n$, $0<\alpha<\infty$, $1\leqslant q\leqslant\infty$, $0<\lambda\leqslant\infty$, $1<p\leqslant\infty$, $n=1,2,\dots$, and let $R(n,p)$ be the class of rational functions $\rho(z)$ of degree $\leqslant n$, analytic for $|z|\leqslant1$, with
\begin{gather*} \|\rho\|_p=\biggl(\,\int_{|\zeta|=1}|\rho(\zeta)|^p\,|d\zeta|\biggr)^{1/p}\leqslant1\\ (\|\rho\|_\infty=\sup\{|\rho(z)|:|z|=1\}). \end{gather*}
It is proved that, if $\alpha\geqslant1+p^{-1}-q^{-1}$, then
$$ \sup\biggl\{\biggl[\,\int_0^1(1-r)^{\alpha\lambda-1}\biggl(\,\int_0^{2\pi}|\rho(r\cdot e^{i\varphi}|^q\,d\varphi\biggr)^{\lambda/q}\,dr\biggr]^{1/\lambda}:\rho\in R(n,p)\biggr\}<\infty. $$

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