On the dependence of the boundary properties of an analytic function on the rapidity of its approximation by rational functions

We investigate the behavior of the means of the modulus of the derivative of an analytic function $f(z)$ which is continuous up to the boundary of its domain $G$, as it depends on the behavior of $R_n(f,\overline G)$, the least deviations of $f$ on $\overline G$ from the rational functions of degree $\leqslant n$. For example, if $p\geqslant1$, $p-1<\alpha\leqslant p$ and $\sum n^{-\alpha+p-1}R_n^p(f,\overline D)<\nobreak\infty$, then $(1-|z|)^{\alpha-1}|f'(z)|^p$ is summable over the area of the disk $D:|z|<1$ (for $p-1<\alpha<p$ this is best possible).
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