Local smoothness of an analytic function compared to the smoothness of its modulus

Vasin, A.; Kislyakov, S.; Medvedev, A.

doi:10.1090/S1061-0022-2014-01296-4

Local smoothness of an analytic function compared to the smoothness of its modulus
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by A. V. Vasin, S. V. Kislyakov and A. N. Medvedev
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 25 (2014), 397-420

DOI: https://doi.org/10.1090/S1061-0022-2014-01296-4

Published electronically: May 16, 2014

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Abstract:

Let $\Phi$ be a function analytic in the disk and continuous up to the boundary, and let its modulus of continuity satisfy the Hölder condition of order $\alpha$, $0<\alpha <2$, at a single boundary point. Under standard assumptions on the zeros of $\Phi$, this function must be then at least $\alpha /2$-Hölder (in a certain integral sense) at the same point. There are generalizations to not necessarily power-type Hölder smoothness.

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