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Transactions of the American Mathematical Society

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

 

 

Grunsky-Nehari inequalities for a subclass of bounded univalent functions


Author: D. W. DeTemple
Journal: Trans. Amer. Math. Soc. 159 (1971), 317-328
MSC: Primary 30.43
DOI: https://doi.org/10.1090/S0002-9947-1971-0279299-7
MathSciNet review: 0279299
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Abstract: Let $ {D_1}$ be the class of regular analytic functions $ F(z)$ in the disc $ U = \{ z:\vert z\vert < 1\} $ for which $ F(0) > 0,\vert F(z)\vert < 1$, and $ F(z) + F(\zeta ) \ne 0$ for all $ z,\zeta \in U$. Inequalities of the Grunsky-Nehari type are obtained for the univalent functions in $ {D_1}$, the proof being based on the area method. By subordination it is shown univalency is unnecessary for certain special cases of the inequalities. Employing a correspondence between $ {D_1}$ and the class $ {S_1}$ of bounded univalent functions, the results can be reinterpreted to apply to this latter class.


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DOI: https://doi.org/10.1090/S0002-9947-1971-0279299-7
Keywords: Univalent functions, Bieberbach-Eilenberg functions, Grunsky inequalities, Nehari inequalities, subordination
Article copyright: © Copyright 1971 American Mathematical Society