Directional convexity of level lines for functions convex in a given direction
HTML articles powered by AMS MathViewer
- by Dmitri V. Prokhorov and Jan Szynal PDF
- Proc. Amer. Math. Soc. 131 (2003), 1453-1457 Request permission
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 $|z|\leq r_{\psi ,\varphi }$ with the property that each function $f\in K(\psi )$ maps this disk onto the convex domain in the direction $e^{i\varphi }$.References
- J. Clunie and T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3–25. MR 752388, DOI 10.5186/aasfm.1984.0905
- A. W. Goodman, Univalent functions. Vol. I, Mariner Publishing Co., Inc., Tampa, FL, 1983. MR 704183
- A. W. Goodman and E. B. Saff, On univalent functions convex in one direction, Proc. Amer. Math. Soc. 73 (1979), no. 2, 183–187. MR 516461, DOI 10.1090/S0002-9939-1979-0516461-2
- Richard Kershner, The number of circles covering a set, Amer. J. Math. 61 (1939), 665–671. MR 43, DOI 10.2307/2371320
- $\{$Collection of articles dedicated to Eberhard Hopf on the occasion of his 70th birthday$\}$, Gordon and Breach, New York-London-Paris, 1973. Applicable Anal. 3 (1973), no. 1. MR 0384367
- D. V. Prokhorov, Level lines of functions that are convex in the direction of an axis, Mat. Zametki 44 (1988), no. 4, 523–527, 558 (Russian); English transl., Math. Notes 44 (1988), no. 3-4, 767–769 (1989). MR 975192, DOI 10.1007/BF01158922
- Stephan Ruscheweyh and Luis C. Salinas, On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 1, 63–73. MR 997971, DOI 10.5186/aasfm.1989.1427
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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1453-1457
- MSC (2000): Primary 30C20; Secondary 30C45
- DOI: https://doi.org/10.1090/S0002-9939-02-06675-3
- MathSciNet review: 1949875