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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Directional convexity of level lines for functions convex in a given direction


Authors: Dmitri V. Prokhorov and Jan Szynal
Journal: Proc. Amer. Math. Soc. 131 (2003), 1453-1457
MSC (2000): Primary 30C20; Secondary 30C45
Published electronically: September 19, 2002
MathSciNet review: 1949875
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Abstract: Let $K(\varphi )$ be the class of functions $f(z)=z+a_{2}z^{2}+\dots $ which are holomorphic and convex in direction $e^{i\varphi }$ in the unit disk $D$, i.e. the domain $f(D)$ is such that the intersection of $f(D)$ and any straight line $\{w:w=w_{0}+te^{i\varphi },t\in \mathbb{R}\}$ is a connected or empty set. In this note we determine the radius $r_{\psi ,\varphi }$ of the biggest disk $\vert z\vert\leq r_{\psi ,\varphi }$ with the property that each function $f\in K(\psi )$ maps this disk onto the convex domain in the direction $e^{i\varphi }$.


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Additional Information

Dmitri V. Prokhorov
Affiliation: Department of Mathematics, Saratov State University, 410026 Saratov, Russia
Email: ProkhorovDV@info.sgu.ru

Jan Szynal
Affiliation: Department of Mathematics, M. Curie-Skłodowska University, 20-031 Lublin, Poland
Email: jsszynal@golem.umcs.lublin.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06675-3
PII: S 0002-9939(02)06675-3
Keywords: Level curves of holomorphic functions, functions convex in one direction
Received by editor(s): August 21, 2001
Received by editor(s) in revised form: December 7, 2001
Published electronically: September 19, 2002
Additional Notes: The first author was partially supported by the RFBR Grant No. 01-01-00123 and the INTAS Grant No. 99-00089
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2002 American Mathematical Society