Composition of Blochs with bounded analytic functions

Abstract:

If $f$ is a holomorphic self-map of the open unit disc and $1 \leq p < \infty$, then the following are equivalent. $(1) h\circ f \in H^{2p}$ for all Bloch functions $h$. \begin{equation*}\underset {{r} }{sup} \int _{0}^{2\pi } \left ( log \frac {1}{1 - \vert f(re^{i\theta })\vert ^{2}}\right )^{p} d\theta < \infty . \tag {2}\end{equation*} \begin{equation*}\int _{0}^{2\pi } \left ( \int _{0}^{1} (f^{\#})^{2}(re^{i\theta }) (1-r) dr \right )^{p} d\theta < \infty , \tag {3}\end{equation*} where $f^{\#}$ is the hyperbolic derivative of $f$: $f^{\#} = \vert f’\vert / (1-\vert f\vert ^{2})$.