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Growth properties and sequences of zeros of analytic functions in spaces of Dirichlet type

Part of: Entire and meromorphic functions, and related topics Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Daniel Girela  and
José Ángel Peláez
Daniel Girela
Affiliation:
Depto. de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain, e-mail: girela@uma.es, pelaez@anamat.cie.uma.es
José Ángel Peláez
Affiliation:
Depto. de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain, e-mail: girela@uma.es, pelaez@anamat.cie.uma.es
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Abstract

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For 0 < p < ∞, we let pp−1 denote the space of those functions f that are analytic in the unit disc Δ = {z ∈ C: |z| < 1} and satisfy ∫Δ(1 – |z|)p−1|f′(z)|pdx dy < ∞ The spaces pp−1 are closely related to Hardy spaces. We have, p−1p ⊂ Hp, if 0 < p ≦ 2, and Hp ⊂ pp−1, if 2 ≦ p < ∞. In this paper we obtain a number of results about the Taylor coefficients of pp-1 -functions and sharp estimates on the growth of the integral means and the radial growth of these functions as well as information on their zero sets.

2000 Mathematics subject classification: primary 30D35, 30D55, 46E15.

Keywords

Spaces of Dirichlet typeHardy spacesBergman spacesintegral meansradial growthsequences of zeros.

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 397 - 418
Copyright
Copyright © Australian Mathematical Society 2006

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