# IJPAM: Volume 21, No. 4 (2005)

NON-STABLE CLASSES OF ANALYTIC FUNCTIONS

Daniel Girela, José Ángel Peláez
Departamento de Análisis Matemático
29071 Málaga, SPAIN
e-mail: girela@uma.es
29071 Málaga, SPAIN
e-mail: pelaez@anamat.cie.uma.es

Abstract.If and are analytic functions in the unit disc , then is said to be weakly subordinate to , written , if there exist analytic functions and , with an inner function, so that . A class of analytic functions in is said to be stable if it is closed under weak subordination, that is, if whenever and are analytic functions in with and . For and , we let denote the weighted Bergman space of all functions , analytic in , such that and the space of Dirichlet type consists of those such that . Among other results, we prove that all the Bergman spaces (, ) and all the -spaces except the space are non-stable classes of analytic functions in .