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ON HARMONIC BLOCH SPACES IN THE UNIT BALL OF ℂn

Part of: Geometric function theory

Published online by Cambridge University Press:  10 June 2011

SH. CHEN  and
X. WANG
SH. CHEN
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, PR China (email: [email protected])
X. WANG*
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, PR China (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball 𝔹n. First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from 𝔹n to ℂ in terms of weighted Lipschitz functions. Then we prove the existence of a Landau–Bloch constant for a class of vector-valued harmonic Bloch mappings from 𝔹n to ℂn.

Keywords

harmonic Bloch spacelittle harmonic Bloch spaceweighted Lipschitz functioncharacterization

MSC classification

Secondary: 30C65: Quasiconformal mappings in $^n$, other generalizations 30C20: Conformal mappings of special domains
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 67 - 78
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

The research was partly supported by NSF of China (No. 11071063), Hunan Provincial Innovation Foundation for Postgraduate (No. 125000-4113) and the Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

References

[1]Arsenović, M., Kojić, V. and Mateljević, M., ‘On Lipschitz continuity of harmonic quasiregular maps on the unit ball in ℝn’, Ann. Acad. Sci. Fenn. Math. 33 (2008), 315–318.Google Scholar
[2]Arsenović, M., Manojlović, V. and Mateljević, M., ‘Lipschitz-type spaces and harmonic mappings in the space’, Ann. Acad. Sci. Fenn. Math. 35 (2010), 379–387.Google Scholar
[3]Chen, Sh., Ponnusamy, S. and Wang, X., ‘Properties of some classes of planar harmonic and planar biharmonic mappings’, Complex Anal. Oper. Theory (2010), doi:10.1007/s11785-010-0061-x.Google Scholar
[4]Chen, Sh., Ponnusamy, S. and Wang, X., ‘Landau’s theorem and Marden constant for harmonic ν-Bloch mappings’, Bull. Aust. Math. Soc. (2011), accepted.CrossRefGoogle Scholar
[5]Chen, Sh., Ponnusamy, S. and Wang, X., ‘Coefficient estimates and Landau’s theorem for planar harmonic mappings’, Bull. Malays. Math. Sci. Soc. 34 (2011), 1–11.Google Scholar
[6]Chen, Sh., Ponnusamy, S. and Wang, X., ‘Bloch and Landau’s theorems for planar p-harmonic mappings’, J. Math. Anal. Appl. 373 (2011), 102–110.Google Scholar
[7]Colonna, F., ‘The Bloch constant of bounded harmonic mappings’, Indiana Univ. Math. J. 38 (1989), 829–840.Google Scholar
[8]Heinz, E., ‘On one-to-one harmonic mappings’, Pacific J. Math. 9 (1959), 101–105.CrossRefGoogle Scholar
[9]Holland, F. and Walsh, D., ‘Criteria for membership of Bloch space and its subspace, BMOA’, Math. Ann. 273 (1986), 317–335.CrossRefGoogle Scholar
[10]Kalaj, D., ‘On the univalent solution of PDE Δu=f between spherical annuli’, J. Math. Anal. Appl. 327 (2007), 1–11.CrossRefGoogle Scholar
[11]Kalaj, D., ‘On harmonic quasiconformal self-mappings of the unit ball’, Ann. Acad. Sci. Fenn. Math. 33 (2008), 261–271.Google Scholar
[12]Kalaj, D., ‘Lipschitz spaces and harmonic mappings’, Ann. Acad. Sci. Fenn. Math. 34 (2009), 475–485.Google Scholar
[13]Li, S. X. and Wulan, H., ‘Characterizations of α-Bloch spaces on the unit ball’, J. Math. Anal. Appl. 337 (2008), 880–887.Google Scholar
[14]Liu, X. Y., ‘Bloch functions of several complex variables’, Pacific J. Math. 152 (1992), 347–363.Google Scholar
[15]Mateljević, M. and Vuorinen, M., ‘On harmonic quasiconformal quasi-isometries’, J. Inequal. Appl. (2010), Article ID 178732, 19 pp.Google Scholar
[16]Rudin, W., Function Theory in the Unit Ball of ℂn (Springer, Berlin, 1980).CrossRefGoogle Scholar
[17]Zhu, K., Spaces of Holomorphic Functions in the Unit Ball (Springer, New York, 2005).Google Scholar