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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Coefficient multipliers of Bloch functions

Authors: J. M. Anderson and A. L. Shields
Journal: Trans. Amer. Math. Soc. 224 (1976), 255-265
MSC: Primary 30A78
MathSciNet review: 0419769
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Abstract: The class $\mathcal {B}$ of Bloch functions is the class of all those analytic functions in the open unit disc for which the maximum modulus is bounded by $c/(1 - r)$ on $|z| \leqslant r$. We study the absolute values of the Taylor coefficients of such functions. In particular, we find all coefficient multipliers from ${l^p}$ into $\mathcal {B}$ and from $\mathcal {B}$ into ${l^p}$. We find the second KΓΆthe dual of $\mathcal {B}$ and show its relevance to the multiplier problem. We identify all power series $\sum {a_n}{z^n}$ such that $\sum {w_n}{a_n}{z^n}$ is a Bloch function for every choice of the bounded sequence $\{ {w_n}\}$. Analogous problems for ${H^p}$ spaces are discussed briefly.

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Article copyright: © Copyright 1976 American Mathematical Society