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

J. Aust. Math. Soc. 80 (2006), 397-418

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

Daniel Girela
Depto. de Análisis Matemático
Facultad de Ciencias
Universidad de Málaga
Campus de Teatinos
29071 Málaga
Spain
girela@uma.es

and

José Ángel Peláez
Depto. de Análisis Matemático
Facultad de Ciencias
Universidad de Málaga
Campus de Teatinos
29071 Málaga
Spain
pelaez@anamat.cie.uma.es

Abstract

For $0<p<\infty$ , we let $\mathcal{D}_{p-1}^p$ denote the space of those functions

that are analytic in the unit disc $\Delta =\{ z\in \mathbb C : | z| <1\}$ and satisfy $\int_\Delta (1-| z|)^{p-1}| f'(z)|^p\, dx\, dy <\infty$ . The spaces $\mathcal{D}_{p-1}^p$ are closely related to Hardy spaces. We have, $\mathcal{D}_{p-1}^p\subset H^p$ , if $0<p\le 2$ , and $H^p\subset \mathcal{D}_{p-1}^p$ , if $2\le p<\infty$ . In this paper we obtain a number of results about the Taylor coefficients of $\mathcal{D}_{p-1}^p$ -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.

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