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Mat. Zametki, 2007, Volume 82, Issue 6, Pages 885–890 (Mi mz4187)  

On the Univalence of Derivatives of Functions which are Univalent in Angular Domains

S. R. Nasyrov

Kazan State University

Abstract: We consider functions $f$ that are univalent in a plane angular domain of angle $\alpha\pi$, $0<\alpha\le2$. It is proved that there exists a natural number $k$ depending only on $\alpha$ such that the $k$th derivatives $f^{(k)}$ of these functions cannot be univalent in this angle. We find the least of the possible values of for $k$. As a consequence, we obtain an answer to the question posed by Kiryatskii: if $f$ is univalent in the half-plane, then its fourth derivative cannot be univalent in this half-plane.

Keywords: univalent function, holomorphic function, Bieberbach's conjecture, Koebe function, Weierstrass theorem

DOI: https://doi.org/10.4213/mzm4187

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English version:
Mathematical Notes, 2007, 82:6, 798–802

Bibliographic databases:

UDC: 517.5
Received: 26.03.2007

Citation: S. R. Nasyrov, “On the Univalence of Derivatives of Functions which are Univalent in Angular Domains”, Mat. Zametki, 82:6 (2007), 885–890; Math. Notes, 82:6 (2007), 798–802

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