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Composition of Blochs with bounded analytic functions


Author: E. G. Kwon
Journal: Proc. Amer. Math. Soc. 124 (1996), 1473-1480
MSC (1991): Primary 30D55, 30D45
DOI: https://doi.org/10.1090/S0002-9939-96-03191-7
MathSciNet review: 1307542
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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 $f$: $f^{\#} = \vert f'\vert / (1-\vert f\vert ^{2})$.


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Additional Information

E. G. Kwon
Affiliation: Department of Mathematics-Education, Andong National University, Andong 760-749, Korea
Email: egkwon@anu.andong.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-96-03191-7
Keywords: $H^{p}$ space, Bloch space, hyperbolic Hardy class, pullbacks
Received by editor(s): January 31, 1994
Received by editor(s) in revised form: October 19, 1994
Additional Notes: This paper was supported by NON DIRECTED RESEARCH FUND, Korea Research Foundation, 1993.
Communicated by: Theodore Gamelin
Article copyright: © Copyright 1996 American Mathematical Society

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