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INVERSE IMAGES OF SECTORS BY FUNCTIONS IN WEIGHTED BERGMAN–ORLICZ SPACES

Part of: Geometric function theory Linear function spaces and their duals

Published online by Cambridge University Press:  01 April 2009

FERNANDO PÉREZ-GONZÁLEZ  and
JULIO C. RAMOS FERNÁNDEZ
FERNANDO PÉREZ-GONZÁLEZ*
Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain (email: fernando.perez.gonzalez@ull.es)
JULIO C. RAMOS FERNÁNDEZ
Affiliation:
Departamento de Matemáticas, Universidad de Oriente, 6101 Cumaná, Edo. Sucre, Venezuela (email: jramos@sucre.udo.edu.ve)
*
✉For correspondence; e-mail: fernando.perez.gonzalez@ull.es
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Abstract

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For ε>0, let Σε={z∈ℂ:∣arg z∣<ε}. It has been proved (D. E. Marshall and W. Smith, Rev. Mat. Iberoamericana15 (1999), 93–116) that ∫ f−1(Σε)∣f(z)∣ dA(z)≃∫ 𝔻∣f(z)∣ dA(z) for every ε>0, uniformly for every univalent function f in the classical Bergman space A1 that fixes the origin. In this paper, we extend this result to those conformal maps on 𝔻 belonging to weighted Bergman–Orlicz classes such that f(0)=∣f′(0)∣−1=0.

Keywords

univalent functionBergman–Orlicz classesmodulus function

MSC classification

Secondary: 30C25: Covering theorems in conformal mapping theory 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 2 , April 2009 , pp. 233 - 247
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The first author has been supported in part by the grant of MEC-Spain MTM2005-07347. Both authors are members of the Spanish Thematic Network MTM2006-26627-E.

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