# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Gel′fer functions, integral means, bounded mean oscillation, and univalencyHTML articles powered by AMS MathViewer

by Shinji Yamashita
Trans. Amer. Math. Soc. 321 (1990), 245-259 Request permission

## Abstract:

A Gelfer function $f$ is a holomorphic function in $D = \{ \left | z \right | < 1\}$ 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 $\{ \left | z \right | = r\}$ 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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