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Transactions of the American Mathematical Society

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

 

 

Families of real and symmetric analytic functions


Authors: Yusuf Abu-Muhanna and Thomas H. MacGregor
Journal: Trans. Amer. Math. Soc. 263 (1981), 59-74
MSC: Primary 30C45; Secondary 30C50
DOI: https://doi.org/10.1090/S0002-9947-1981-0590411-9
MathSciNet review: 590411
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Abstract: We introduce families of functions analytic in the unit disk and having rotational symmetries. The families include the $ k$-fold symmetric univalent functions which have real coefficients. We relate the families to special classes of functions with a positive real part and then determine their extreme points. The case $ k = 2$ corresponds to the odd functions which "preserve quadrants" and the extreme points of this set are characterized by having a radial limit which is real or imaginary almost everywhere. We also find estimates on the initial coefficients of functions in the families.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0590411-9
Keywords: Analytic function, typically-real function, extreme point, univalent function, functions with a positive real part, radial limit function, Poisson representation, harmonic measure, subordination, $ k$-fold symmetric function, coefficient estimates
Article copyright: © Copyright 1981 American Mathematical Society