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Local smoothness of an analytic function compared to the smoothness of its modulus. (English. Russian original) Zbl 1302.30075

St. Petersbg. Math. J. 25, No. 3, 397-420 (2014); translation from Algebra Anal. 25, No. 3, 52-85 (2013).
Summary: 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.

MSC:

30J99 Function theory on the disc
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