On univalent functions convex in one direction
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- by A. W. Goodman and E. B. Saff PDF
- Proc. Amer. Math. Soc. 73 (1979), 183-187 Request permission
Abstract:
Let $f(z) = z + \Sigma _2^\infty {a_k}{z^k}$ be analytic and univalent in the unit disk $E:|z| < 1$ and map the disk onto a domain which is convex in the direction of the imaginary axis. We show by example that for $\sqrt 2 - 1 < r < 1$, the function $f(z)$ need not map the disk $|z| < r$ onto a domain convex in the direction of the imaginary axis. We also find the largest domain contained in $f(E)$ for every normalized $f(z)$ that maps E onto a domain convex in the direction of the imaginary axis.References
- Walter Hengartner and Glenn Schober, On Schlicht mappings to domains convex in one direction, Comment. Math. Helv. 45 (1970), 303–314. MR 277703, DOI 10.1007/BF02567334
- W. Hengartner and G. Schober, A remark on level curves for domains convex in one direction, Applicable Anal. 3 (1973), 101–106. MR 393450, DOI 10.1080/00036817308839059
- M. S. Robertson, Analytic Functions Star-Like in One Direction, Amer. J. Math. 58 (1936), no. 3, 465–472. MR 1507169, DOI 10.2307/2370963
- W. C. Royster and Michael Ziegler, Univalent functions convex in one direction, Publ. Math. Debrecen 23 (1976), no. 3-4, 339–345. MR 425101
Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 73 (1979), 183-187
- MSC: Primary 30C45
- DOI: https://doi.org/10.1090/S0002-9939-1979-0516461-2
- MathSciNet review: 516461