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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On univalent functions convex in one direction


Authors: A. W. Goodman and E. B. Saff
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
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Abstract: Let $ f(z) = z + \Sigma _2^\infty {a_k}{z^k}$ be analytic and univalent in the unit disk $ E:\vert z\vert < 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 $ \vert z\vert < 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.


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DOI: https://doi.org/10.1090/S0002-9939-1979-0516461-2
Article copyright: © Copyright 1979 American Mathematical Society

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