Geodesic
On the dependence of the boundary properties of an analytic function on the rapidity of its approximation by rational functions
Sbornik. Mathematics, Tome 32 (1977) no. 1, pp. 116-126 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 this is best possible). Bibliography: 6 titles. 
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E. P. Dolzhenko. On the dependence of the boundary properties of an analytic function on the rapidity of its approximation by rational functions. Sbornik. Mathematics, Tome 32 (1977) no. 1, pp. 116-126. http://geodesic.mathdoc.fr/item/SM_1977_32_1_a7/

[1] E. P. Dolzhenko, “Ratsionalnye approksimatsii i granichnye svoistva analiticheskikh funktsii”, Matem. sb., 69 (111) (1966), 497–524 | Zbl

[2] E. A. Sevastyanov, “Kusochno monotonnaya approksimatsiya i $\Phi$-variatsii”, Analysis Math., 1:2 (1975), 141–164 | DOI | MR

[3] E. A. Sevastyanov, “Ravnomernye priblizheniya kusochno monotonnymi funktsiyami i nekotorye ikh prilozheniya k $\Phi$-variatsiyam i ryadam Fure”, DAN SSSR, 217:1 (1974), 27–30

[4] E. P. Dolzhenko, “Ravnomernye approksimatsii ratsionalnymi funktsiyami (algebraicheskimi i trigonometricheskimi) i globalnye funktsionalnye svoistva”, DAN SSSR, 166:3 (1966), 526–529 | Zbl

[5] O. V. Besov, V. P. Ilin, S. M. Nikolskii, Integralnye predstavleniya funktsii i teoremy vlozheniya, izd-vo «Nauka», Moskva, 1975 | MR | Zbl

[6] M. I. Gvaradze, “O prostranstvakh $B(p, q, \lambda)$ analiticheskikh funktsii”, Soobsch. AN Gruz. SSR, 77:2 (1975), 273–276 | MR | Zbl