Gel′fer functions, integral means, bounded mean oscillation, and univalency

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.