A characterization of random Bloch functions. (English) Zbl 0973.30027
A Bloch function is an analytic function \(f\) on the unit disk \(D=\{z:|z|<1\}\) such that \(\sup_{z\in D}(1-|z|^2)|f'(z)|<\infty\). It is known that the set of all Bloch functions becomes a Banach space (called the Bloch space) with respect to the norm \(\|f\|_B=|f(0)|+\sup_{z\in D}(1-|z|^2)|f'(z)|\).
In this paper the author obtains a necessary and sufficient condition for the complex sequence \(\{a_n\}\) with \(\sum|a_n|^2<\infty\) such that \(\sum_{n=1}^\infty\pm a_nz^n\) is a Bloch function for almost all choices of signs \(\pm\). Also the author answers to a question of J. M. Anderson, J. Clunie and C. Pommerenke [J. Reine Angew. Math. 270, 12-37 (1974; Zbl 0292.30030)].
In this paper the author obtains a necessary and sufficient condition for the complex sequence \(\{a_n\}\) with \(\sum|a_n|^2<\infty\) such that \(\sum_{n=1}^\infty\pm a_nz^n\) is a Bloch function for almost all choices of signs \(\pm\). Also the author answers to a question of J. M. Anderson, J. Clunie and C. Pommerenke [J. Reine Angew. Math. 270, 12-37 (1974; Zbl 0292.30030)].
Reviewer: Gabriela Kohr (Cluj-Napoca)
MSC:
| 30D45 | Normal functions of one complex variable, normal families |
Citations:
Zbl 0292.30030References:
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| [2] | Anderson, J. M., Random power series, Lecture Notes in Math., 1573, 174-174 (1994) |
| [3] | Anderson, J. M.; Clunie, J.; Pommerenke, Ch., On Bloch functions and normal functions, J. Reine Angew. Math., 270, 12-37 (1974) · Zbl 0292.30030 |
| [4] | Duren, P., Random series and bounded mean oscillation, Michigan Math. J., 32, 81-86 (1985) · Zbl 0565.30001 |
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