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Proceedings of the American Mathematical Society

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



Radii problems for sections of convex functions

Author: Herb Silverman
Journal: Proc. Amer. Math. Soc. 104 (1988), 1191-1196
MSC: Primary 30C45; Secondary 30C50
MathSciNet review: 942638
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Abstract: A classical theorem of Szegö states that the sections $ {f_n}(z) = z + \sum\nolimits_{k = 2}^n {{a_k}{z^k}} $ of a convex function $ f(z) = z + \sum\nolimits_{k = 2}^\infty {{a_k}{z^k}} $ must be convex for $ \left\vert z \right\vert < \frac{1}{4}$. We determine disks $ \left\vert z \right\vert < {r_n}$ in which $ {f_n}$ is starlike and starlike of a positve order. Our proofs rely on some properties of convolutions.

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Keywords: Univalent, starlike, convex, convolution
Article copyright: © Copyright 1988 American Mathematical Society

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