Integral means, bounded mean oscillation, and Gel′fer functions

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 $\log f,f \in G$.