IJPAM: Volume 21, No. 4 (2005)

NON-STABLE CLASSES OF ANALYTIC FUNCTIONS

Daniel Girela

, José Ángel Peláez

Departamento de Análisis Matemático
Facultad de Ciencias
Universidad de Málaga
29071 Málaga, SPAIN
e-mail: girela@uma.es

Departamento de Matemática Aplicada
Universidad de Málaga
29071 Málaga, SPAIN
e-mail: pelaez@anamat.cie.uma.es

Abstract.If and are analytic functions in the unit disc $\mathbb D$ , then is said to be weakly subordinate to , written $f\prec^w g$ , if there exist analytic functions and $\om:\mathbb D\to\mathbb D$ , with an inner function, so that $f\circ\f=g\circ\om$ . A class of analytic functions in $\mathbb D$ is said to be stable if it is closed under weak subordination, that is, if $\,f\in X$ whenever $\,f$ and $\,g$ are analytic functions in $\mathbb D$ with $g\in X$ and $f\prec^w g$ . For $0<p<\infty$ and $\alpha >-1$ , we let $A^p_\alpha$ denote the weighted Bergman space of all functions , analytic in $\mathbb D$ , such that $f\in L^p\left ((1-\vert z\vert ^2)^\alpha dxdy\right )$ and the space of Dirichlet type $\Dpa$ consists of those such that $f'\in A^p_\alpha$ . Among other results, we prove that all the Bergman spaces $A^p_\alpha$ ( $0<p<\infty$ , $\alpha >-1$ ) and all the $\Dpa$ -spaces except the space $\mathcal D^2_1=H^2$ are non-stable classes of analytic functions in $\mathbb D$ .