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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.

References

[1]Castillo, R. and Ramos Fernández, J., ‘Angular distribution of mass by Besov functions’, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), 303–310.Google Scholar
[2]Duren, P. L. and Schuster, A., Bergman Spaces (American Mathematical Society, Providence, RI, 2004).CrossRefGoogle Scholar
[3]Garnett, J. B. and Marshall, D. E., Harmonic Measure (Cambridge University Press, New York, 2005).CrossRefGoogle Scholar
[4]Gehring, F. and Palka, B., ‘Quasiconformally homogeneous domains’, J. Anal. Math. 30 (1976), 172–199.CrossRefGoogle Scholar
[5]He, Y., ‘B a spaces and Orlicz spaces’, Function Spaces and Complex Analysis, Report Series, 2, Department of Mathematics, University of Joensuu, 1999, pp. 37–62.Google Scholar
[6]Hedenmalm, H., Korenblum, B. and Zhu, K., Theory of Bergman Spaces (Springer, New York, 2000).CrossRefGoogle Scholar
[7]Marshall, D. E. and Smith, W., ‘The angular distribution of mass by Bergman functions’, Rev. Mat. Iberoamericana 15 (1999), 93–116.CrossRefGoogle Scholar
[8]Mazur, S. and Orlicz, W., ‘On some classes of linear spaces’, Studia Math. 17 (1958), 97–119.CrossRefGoogle Scholar
[9]Ortel, M. and Smith, W., ‘The argument of an extremal dilatation’, Proc. Amer. Math. Soc. 104 (1988), 498–502.CrossRefGoogle Scholar
[10]Pérez-González, F. and Ramos Fernández, J., ‘The angular distribution of mass by weighted Bergman functions’, Divulg. Mat. 12 (2004), 65–86.Google Scholar
[11]Pérez-González, F. and Ramos Fernández, J., ‘On dominating sets for Bergman space’, Contemp. Math. 404 (2006), 175–186.CrossRefGoogle Scholar
[12]Pommerenke, Ch., Boundary Behavior of Conformal Maps (Springer, Berlin, 1992).CrossRefGoogle Scholar
[13]Ramos Fernández, J., ‘Supremum over inverse image of functions in the Bloch space’, C. R. Acad. Sci. Paris, Ser. I 344 (2007), 291–294.CrossRefGoogle Scholar
[14]Rao, M. M. and Ren, Z. D., Theory of Orlicz Spaces, Pure and Applied Mathematics, 146 (Marcel Dekker, New York, 1991).Google Scholar
[15]Sharma, S., Jagdish, R. and Renu, A., ‘Composition operators on Bergman–Orlicz type spaces’, Indian J. Math. 40 (1998), 227–235.Google Scholar
[16]Stević, S., ‘On generalized weighted Bergman space’, Complex Var. Theory Appl. 49 (2004), 109–124.Google Scholar
[17]Wang, X. and Xu, A., ‘Orlicz–Bergman space and their composition operators’, Sichuan Daxue Xuebao 40 (2003), 24–28.Google Scholar
[18]Zhu, K., Operator Theory in Function Spaces, Pure and Applied Mathematics, 139 (Marcel Dekker, New York, 1990).Google Scholar