Composition of Blochs with bounded analytic functions
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- by E. G. Kwon PDF
- Proc. Amer. Math. Soc. 124 (1996), 1473-1480 Request permission
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 $f$: $f^{\#} = \vert fâ\vert / (1-\vert f\vert ^{2})$.References
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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
- 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
- © Copyright 1996 American Mathematical Society
- 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