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

Authors: A. V. Vasin, S. V. Kislyakov and A. N. Medvedev
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 25 (2013), nomer 3.
Journal: St. Petersburg Math. J. 25 (2014), 397-420
MSC (2010): Primary 30H25
Published electronically: May 16, 2014
MathSciNet review: 3184598
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Abstract | References | Similar Articles | Additional Information

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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Additional Information

A. V. Vasin
Affiliation: State University of Maritime and Inland Shipping, ul. Dvinskaya 5/7, St. Petersburg 158035, Russia

S. V. Kislyakov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

A. N. Medvedev
Affiliation: St. Petersburg State University, Universitetskii pr. 28, St. Petersburg 198504, Russia

Keywords: Outer function, harmonic conjugation operator, mean oscillation, finite differences
Received by editor(s): February 28, 2013
Published electronically: May 16, 2014
Additional Notes: Supported by RFBR (the first and the second authors), grant 11-01-00526
Dedicated: Dedicated to Boris Mikhaĭlovich Makarov
Article copyright: © Copyright 2014 American Mathematical Society