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Integral means, bounded mean oscillation, and Gelfer functions


Author: Daniel Girela
Journal: Proc. Amer. Math. Soc. 113 (1991), 365-370
MSC: Primary 30C80; Secondary 30D50
DOI: https://doi.org/10.1090/S0002-9939-1991-1065948-0
MathSciNet review: 1065948
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Abstract: A Gelfer function $ f$ is a holomorphic function in the unit disc $ D = \{ z:\vert z\vert < 1\} $ such that $ f(0) = 1$ and $ f(z) + f(w) \ne 0$ 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$. Yamashita has recently proved that if $ f$ is a Gelfer function then $ f \in {H^p},0 < p < 1$ while $ \log f \in \operatorname{BMOA}$ and $ {\left\Vert {\log f} \right\Vert _{\operatorname{BMO}{{\text{A}}_2}}} \leq \pi /\sqrt 2 $. In this paper we prove that the function $ \lambda (z) = (1 + z)/(1 - z)$ is extremal for a very large class of problems about integral means in the class $ G$. This result in particular implies that $ G \subset {H^p},0 < p < 1$, and we use it also to obtain a new proof of a generalization of Yamashita's estimation of the BMOA norm of $ \log f,f \in G$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1991-1065948-0
Keywords: Integral means, bounded mean oscillation, Gelfer functions, symmetrization
Article copyright: © Copyright 1991 American Mathematical Society

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