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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On certain convex sets in the space of locally schlicht functions
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by Y. J. Kim and E. P. Merkes
Trans. Amer. Math. Soc. 196 (1974), 217-224
DOI: https://doi.org/10.1090/S0002-9947-1974-0349981-4

Abstract:

Let $H = H{(^ \ast },[ + ])$ denote the real linear space of locally schlicht normalized functions in $|z| < 1$ 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.
References
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 196 (1974), 217-224
  • MSC: Primary 30A32; Secondary 30A98
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0349981-4
  • MathSciNet review: 0349981