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Transactions of the American Mathematical Society

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

 

 

A generalization of univalent functions with bounded boundary rotation


Author: Edward J. Moulis
Journal: Trans. Amer. Math. Soc. 174 (1972), 369-381
MSC: Primary 30A32
DOI: https://doi.org/10.1090/S0002-9947-1972-0320296-1
MathSciNet review: 0320296
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Abstract: This paper introduces a class of functions which generalizes both those functions $ f(z)$ with bounded boundary rotation and those functions for which $ zf'(z)$ is a-spirallike. A simple variational formula for this class is derived and used to determine sufficient conditions for the univalency of functions there in. Various representations for these functions are given, and these are used to derive another condition for univalence; this one is the best known so far in the subclass consisting of functions $ f(z)$ for which $ zf'(z)$ is a-spirallike. Bounds on the modulus of the Schwarzian derivative are also derived; these are sharp in the subclass of functions having bounded boundary rotation.


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DOI: https://doi.org/10.1090/S0002-9947-1972-0320296-1
Keywords: Univalent functions, boundary rotation, a-spirillike functions, Schwarzian derivative
Article copyright: © Copyright 1972 American Mathematical Society