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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



On means with power $ -2$ for the dervatives of functions of class $ S$

Author: N. A. Shirokov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 28 (2016), nomer 6.
Journal: St. Petersburg Math. J. 28 (2017), 855-867
MSC (2010): Primary 30C75
Published electronically: October 2, 2017
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Abstract: Let $ S$ be the standard class of conformal mapping of the unit disk $ \mathbb{D}$, and let $ F\in \mathbb{D}$. Suppose that there exist Jordan domains $ G_1$ and $ G$, $ G_1\supset G$, such that $ G\subset \mathbb{C}\setminus f(\mathbb{D})$, $ \partial f(\mathbb{D})\cap \partial G$ contains a Dini-smoth arc $ \gamma $, and $ G_1 \cap \partial f(\mathbb{D}) \cap \partial G=\gamma $. It is established that, in this case, for any $ r$ with $ 0<r<1$, $ F$ does not maximize the expression

$\displaystyle \int _{\vert z\vert=r}\frac {1}{\vert F'(z)\vert^2} \,\vert dz\vert$    

in the class $ S$.

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Additional Information

N. A. Shirokov
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State university, Universitetskiĭ pr. 28, Petrodvorets, 198504 St. Petersburg, Russia; Research institute Higher School of Economics, Soyuza Pechatnikov 16, St. Petersburg, Russia

Keywords: Brennan's conjecture, conformal mappings, means of the derivative of a conformal mapping, the class $S$.
Received by editor(s): June 7, 2016
Published electronically: October 2, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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