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On the zeros of Bloch functions

Published online by Cambridge University Press:  01 July 2000

DANIEL GIRELA ,
MARIA NOWAK  and
PIOTR WANIURSKI
DANIEL GIRELA
Affiliation:
Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain; e-mail: girela@anamat.cie.uma.es
MARIA NOWAK
Affiliation:
Instytut Matematyki UMCS, Plac M. Curie-Sklodowskiej 1, 20-301 Lublin, Poland; e-mail: nowakm@golem.umcs.lublin.pl
PIOTR WANIURSKI
Affiliation:
Instytut Matematyki UMCS, Plac M. Curie-Sklodowskiej 1, 20-301 Lublin, Poland; e-mail: peter@golem.umcs.lublin.pl
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Abstract

A function f, analytic in the unit disc Δ, is said to be a Bloch function if supz∈Δ(1 − [mid ]z[mid ]2)[mid ]f′(z)[mid ] < ∞. In this paper we study the zero sequences of non-trivial Bloch functions. Among other results we prove that if f is a Bloch function with f(0) ≠ 0 and {zk} is the sequence of ordered zeros of f. then

formula here

and

formula here

We will also prove that (ii) is best possible even for the little Bloch space [Bscr ]0. To this end we construct a function f ∈ [Bscr ]0 whose zero sequence {zk} satisfies

formula here

We also consider analogous problems for some other related spaces of analytic functions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 129 , Issue 1 , July 2000 , pp. 117 - 128
Copyright
2000 Cambridge Philosophical Society

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