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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Gelfer functions, integral means, bounded mean oscillation, and univalency


Author: Shinji Yamashita
Journal: Trans. Amer. Math. Soc. 321 (1990), 245-259
MSC: Primary 30C45; Secondary 30C50
MathSciNet review: 1010891
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Abstract: A Gelfer function $ f$ is a holomorphic function in $ D = \{ \left\vert z \right\vert < 1\} $ such that $ f(0) = 1$ and $ f(z) \ne - f(w)$ for all $ z$, $ w$ in $ D$. The family $ G$ of Gelfer functions contains the family $ P$ of holomorphic functions $ f$ in $ D$ with $ f(0) = 1$ and Re $ f > 0$ in $ D$. If $ f$ is holomorphic in $ D$ and if the $ {L^2}$ mean of $ f'$ on the circle $ \{ \left\vert z \right\vert = r\} $ is dominated by that of a function of $ G$ as $ r \to 1 - 0$, then $ f \in BMOA$. This has two recent and seemingly different results as corollaries. A core of the proof is the fact that $ {\operatorname{log}}f \in BMOA$ if $ f \in G$. Besides the properties obtained concerning $ f \in G$ itself, we shall investigate some families of functions where the roles played by $ P$ in Univalent Function Theory are replaced by those of $ G$. Some exact estimates are obtained.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1990-1010891-X
PII: S 0002-9947(1990)1010891-X
Article copyright: © Copyright 1990 American Mathematical Society