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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On certain convex sets in the space of locally schlicht functions

Authors: Y. J. Kim and E. P. Merkes
Journal: Trans. Amer. Math. Soc. 196 (1974), 217-224
MSC: Primary 30A32; Secondary 30A98
MathSciNet review: 0349981
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Abstract: Let $ H = H{(^ \ast },[ + ])$ denote the real linear space of locally schlicht normalized functions in $ \vert z\vert < 1$ as defined by Hornich. Let K and C respectively be the classes of convex functions and the close-to-convex functions. If $ M \subset H$ there is a closed nonempty convex set in the $ \alpha \beta $-plane such that for $ (\alpha ,\beta )$ in this set $ {\alpha ^ \ast }f[ + ]{\beta ^ \ast }g \in C$ (in K) whenever f, $ g \in M$. This planar convex set is explicitly given when M is the class K, the class C, and for other classes. Some consequences of these results are that K and C are convex sets in H and that the extreme points of C are not in K.

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Keywords: Univalent, convex close-to-convex, extreme point
Article copyright: © Copyright 1974 American Mathematical Society