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Proceedings of the American Mathematical Society

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

 

 

On the coefficients of starlike functions


Author: Finbarr Holland
Journal: Proc. Amer. Math. Soc. 33 (1972), 463-470
MSC: Primary 30A34
MathSciNet review: 0291438
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Abstract: Every probability measure $ \mu $ on the circle group generates a function f that is starlike univalent on the open unit disc $ \Delta $. In this note the relationship between $ ({c_n})$, the Fourier-Stieltjes coefficients of $ \mu $, and $ ({a_n})$, the Taylor coefficients of f, is examined. A number of theroems are presented which indicate (possibly in the presence of fairly mild restrictions) that the sequences $ ({c_n})$ and $ (n{a_n})$ behave similarly. For example, it is shown that if $ f(\Delta )$ is finite, then $ ({c_n})$ converges to zero if, and only if, $ (n{a_n})$ converges to zero, thereby completing a result of Pommerenke.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0291438-7
Keywords: Starlike function, univalent function, domain of finite area, probability measure, Banach space, closed linear span
Article copyright: © Copyright 1972 American Mathematical Society