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Proceedings of the American Mathematical Society

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

 

 

A generalization of Study's theorem on convex maps


Author: Roger W. Barnard
Journal: Proc. Amer. Math. Soc. 72 (1978), 127-134
MSC: Primary 30C45
MathSciNet review: 503546
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Abstract: Study proved that disks of radius r, $ 0 < r \leqslant 1$, get mapped onto convex (starlike) sets under all convex (starlike) maps of the unit disk U.

Pommerenke and Heins gave a geometric characterization of the subsets of U that get mapped onto convex sets under all convex maps. In this note we give an analytical characterization of the subsets of U that get mapped onto $ \alpha $-starlike domains under all $ \alpha $-starlike maps of U for all $ \alpha \geqslant 0$ noting that 0-starlikeness equals starlikeness and 1-starlikeness equals convexity.


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DOI: https://doi.org/10.1090/S0002-9939-1978-0503546-9
Keywords: $ \alpha $-starlikeness, Study's Theorem
Article copyright: © Copyright 1978 American Mathematical Society