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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Integral means, bounded mean oscillation, and Gel′fer functions
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by Daniel Girela PDF
Proc. Amer. Math. Soc. 113 (1991), 365-370 Request permission

Abstract:

A Gelfer function $f$ is a holomorphic function in the unit disc $D = \{ z:|z| < 1\}$ 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 ${\left \| {\log f} \right \|_{\operatorname {BMO}{{\text {A}}_2}}} \leq \pi /\sqrt 2$. 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
  • © Copyright 1991 American Mathematical Society
  • 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