On means with power $-2$ for the dervatives of functions of class $S$
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N. A. Shirokov
Translated by: A. Plotkin - St. Petersburg Math. J. 28 (2017), 855-867
- DOI: https://doi.org/10.1090/spmj/1477
- 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 \begin{equation*} \int _{|z|=r}\frac {1}{|F’(z)|^2} |dz| \end{equation*} in the class $S$.References
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Bibliographic Information
- N. A. Shirokov
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State university, UniversitetskiÄ pr. 28, Petrodvorets, 198504 St. Petersburg, Russia; Research institute Higher School of Economics, Soyuza Pechatnikov 16, St. Petersburg, Russia
- Email: nikolai.shirokov@gmail.com
- Received by editor(s): June 7, 2016
- Published electronically: October 2, 2017
- © Copyright 2017 American Mathematical Society
- Journal: St. Petersburg Math. J. 28 (2017), 855-867
- MSC (2010): Primary 30C75
- DOI: https://doi.org/10.1090/spmj/1477
- MathSciNet review: 3637581