Title: Mean growth of the derivative of analytic functions, bounded mean oscillation, and normal functions

Authors: Oscar Blasco, Daniel Girela and M. Auxiliadora Marquez

Issue: Volume 47 (1998), Issue 3, 893-912

Abstract: For a given positive function $\varphi$ defined in $[0,1)$ and $1 \leq p < \infty$, we consider the space $\mathcal{L}(p,\varphi)$ which consists of all functions $f$ analytic in the unit disk $\Delta$ for which \[ \left(\frac{1}{2\pi} \int_{-\pi}^{\pi} |f'(re^{i\theta}|^p \,\mathrm{d}\theta\right)^{1/\pi} = \mathrm{O}(\varphi(r)),\quad\text{as }r \to 1. \] A result of Bourdon, Shapiro and Sledd implies that such a space is contained in BMOA for $\varphi(r) = (1 - r)^{1/p - 1}$. Among other results, in this paper we prove that this result is sharp in a very strong sense, showing that, for a large class of weight functions $\varphi$, the function $\varphi(r) = (1 - r)^{1/p - 1}$ is the best one to get $\mathcal{L}(p,\varphi) \in \mathrm{BMOA}$. Actually, if $\varphi(r)(1 - r)^{1 - 1/p} \uparrow \infty$. as $r \uparrow 1$, we construct a function $f \in \mathcal{L}(p,\varphi)$ which is not a normal function. These results improve other results obtained recently by the second author. We also characterize the functions $\varphi$, among a certain class of weight functions, to be able to embed $\mathcal{L}(p,\varphi)$ into $H^q$ for $q > p$ or into the space $\mathcal{B}$ of Bloch functions.