Proceedings of the American Mathematical Society

Author(s): E. G. Kwon
Journal: Proc. Amer. Math. Soc. 124 (1996), 1473-1480.
MSC (1991): Primary 30D55, 30D45
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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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Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D55, 30D45

E.G. Kwon, On analytic functions of Bergman BMO in the ball, Canad. Math. Bull. 42 (1999), 97-103.

Ruhan Zhao, Composition Operators from Bloch Type Spaces to Hardy and Besov Spaces, Journal of Mathematical Analysis and Applications 233 (1999), 749-766.

F. Perez-Gonzalez and J. Xiao, Bloch-Hardy pullbacks, Acta Sci. Math. 67 (2001), 709-718 .

E.G. Kwon, Hyperbolic mean growth of bounded holomorphic functions in the ball, Trans. Amer. Math. Soc. 355 (2003), 1269-1294.

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