IJPAM: Volume 21, No. 4 (2005)


Daniel Girela$^1$, José Ángel Peláez$^2$
$^1$Departamento de Análisis Matemático
Facultad de Ciencias
Universidad de Málaga
29071 Málaga, SPAIN
e-mail: girela@uma.es
$^2$Departamento de Matemática Aplicada
Universidad de Málaga
29071 Málaga, SPAIN
e-mail: pelaez@anamat.cie.uma.es

Abstract.If $f$ and $g$ are analytic functions in the unit disc $\mathbb D$, then $f$ is said to be weakly subordinate to $g$, written $f\prec^w g$, if there exist analytic functions $f$ and $\om:\mathbb D\to\mathbb D$, with $f$ an inner function, so that $f\circ\f=g\circ\om$. A class $X$ 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 $f$, 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 $f$ 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 $\mathbb D$.

Received: May 25, 2005

AMS Subject Classification: 30D45, 30D50, 30D55

Key Words and Phrases: weak subordination, stable class, Hardy spaces, Dirichlet spaces, Bloch function, inner function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 21
Issue: 4