Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T14:39:00.376Z Has data issue: false hasContentIssue false
  • English
  • Français

COEFFICIENT ESTIMATES, LANDAU’S THEOREM AND LIPSCHITZ-TYPE SPACES ON PLANAR HARMONIC MAPPINGS

Part of: Other generalizations Geometric function theory Spaces and algebras of analytic functions

Published online by Cambridge University Press:  01 April 2014

SHAOLIN CHEN ,
SAMINATHAN PONNUSAMY  and
ANTTI RASILA
SHAOLIN CHEN
Affiliation:
Department of Mathematics and Computational Science, Hengyang Normal University, Hengyang, Hunan 421008,PR China email mathechen@126.com
SAMINATHAN PONNUSAMY*
Affiliation:
Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and Security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India email samy@isichennai.res.insamy@iitm.ac.in
ANTTI RASILA
Affiliation:
Department of Mathematics and Systems Analysis, Aalto University, PO Box 11100, FI-00076 Aalto, Finland email antti.rasila@iki.fi
*
✉
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we investigate the properties of locally univalent and multivalent planar harmonic mappings. First, we discuss coefficient estimates and Landau’s theorem for some classes of locally univalent harmonic mappings, and then we study some Lipschitz-type spaces for locally univalent and multivalent harmonic mappings.

Keywords

harmonic mappingsharmonic univalent mappingsLipschitz spaceharmonic Bloch spaceGreen’s theorem

MSC classification

Secondary: 30H05: Bounded analytic functions 30H10: Hardy spaces 30H30: Bloch spaces 30C20: Conformal mappings of special domains 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.) 30C62: Quasiconformal mappings in the plane 31C05: Harmonic, subharmonic, superharmonic functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 96 , Issue 2 , April 2014 , pp. 198 - 215
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

References

Arsenović, M., Manojlović, V. and Mateljević, M., ‘Lipschitz-type spaces and quasiregular harmonic mappings in the space’, Ann. Acad. Sci. Fenn. Math. 35 (2010), 379–387.Google Scholar
Chen, H., Gauthier, P. M. and Hengartner, W., ‘Bloch constants for planar harmonic mappings’, Proc. Amer. Math. Soc. 128 (2000), 3231–3240.Google Scholar
Chen, H. and Gauthier, P. M., ‘The Landau theorem and Bloch theorem for planar harmonic and pluriharmonic mappings’, Proc. Amer. Math. Soc. 139 (2011), 583–595.Google Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Bloch constant and Landau’s theorems for planar p-harmonic mappings’, J. Math. Anal. Appl. 373 (2011), 102–110.Google Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Landau’s theorem and Marden constant for harmonic $\nu $-Bloch mappings’, Bull. Aust. Math. Soc. 84 (2011), 19–32.Google Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Properties of some classes of planar harmonic and planar biharmonic mappings’, Complex Anal. Oper. Theory 5 (2011), 901–916.Google Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Coefficient estimates and Landau-Bloch’s constant for planar harmonic mappings’, Bull. Malays. Math. Sci. Soc. 34 (2011), 255–265.Google Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘On planar harmonic Lipschitz and planar harmonic Hardy classes’, Ann. Acad. Sci. Fenn. Math. 36 (2011), 567–576.CrossRefGoogle Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Equivalent moduli of continuity, Bloch’s theorem for pluriharmonic mappings in ${ \mathbb{B} }^{n} $’, Proc. Indian Acad. Sci. Math. Sci. 122 (2012), 583–595.CrossRefGoogle Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Weighted Lipschitz continuity, Schwarz-Pick’s lemma and Landau-Bloch’s theorem for hyperbolic harmonic functions in the unit ball’, Math. Model. Anal. 18 (2013), 66–79.Google Scholar
Chen, Sh., Ponnusamy, S. and Wang, X., ‘Harmonic mappings in Bergman spaces’, Monatsh. Math. 170 (2013), 325–342.Google Scholar
Chuaqui, M., Duren, P. and Osgood, B., ‘Curvature properties of planar harmonic mappings’, Comput. Methods Funct. Theory 4 (2004), 127–142.CrossRefGoogle Scholar
Chuaqui, M. and Hernández, R., ‘Univalent harmonic mappings and linearly connected domains’, J. Math. Anal. Appl. 332 (2007), 1189–1194.Google Scholar
Clunie, J. G. and Sheil-Small, T., ‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 3–25.Google Scholar
Colonna, F., ‘The Bloch constant of bounded harmonic mappings’, Indiana Univ. Math. J. 38 (1989), 829–840.Google Scholar
Dorff, M. and Nowak, M., ‘Landau’s theorem for planar harmonic mappings’, Comput. Methods Funct. Theory 4 (2004), 151–158.Google Scholar
Duren, P., Harmonic Mappings in the Plane (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
Dyakonov, K. M., ‘Equivalent norms on Lipschitz-type spaces of holomorphic functions’, Acta Math. 178 (1997), 143–167.Google Scholar
Dyakonov, K. M., ‘Holomorphic functions and quasiconformal mappings with smooth moduli’, Adv. Math. 187 (2004), 146–172.CrossRefGoogle Scholar
Grigoryan, A., ‘Landau and Bloch theorems for planar harmonic mappings’, Complex Var. Elliptic Equ. 51 (2006), 81–87.Google Scholar
Koh, N. T. and Kovalev, L. V., ‘Area constraction for harmonic automorphisms of the disk’, Bull. Lond. Math. Soc. 43 (2011), 91–96.Google Scholar
Korenblum, B., ‘BMO estimates and radial growth of Bloch functions’, Bull. Amer. Math. Soc. 12 (1985), 99–102.Google Scholar
Lewy, H., ‘On the non-vanishing of the Jacobian in certain one-to-one mappings’, Bull. Amer. Math. Soc. 42 (1936), 689–692.CrossRefGoogle Scholar
Liu, M., ‘Estimates on Bloch constants for planar harmonic mappings’, Sci. China Ser. A Math. 52 (1) (2009), 87–93.Google Scholar
Mateljević, M., ‘Quasiconformal and quasiregular harmonic analogues of Koebe’s theorem and applications’, Ann. Acad. Sci. Fenn. Math. 32 (2007), 301–315.Google Scholar
Mateljević, M. and Vuorinen, M., ‘On harmonic quasiconformal quasiisometries’, J. Inequal. Appl. (2010), Article ID 178732, 19 pages, doi:10.1155/2010/1787.Google Scholar
Nehari, Z., Conformal Mapping, Reprinting of the 1952 edition (Dover Publications, New York, 1975).Google Scholar
Pavlović, M., ‘On Dyakonov’s paper “Equivalent norms on Lipschitz-type spaces of holomorphic functions”’, Acta Math. 183 (1999), 141–143.Google Scholar
Pavlović, M., Introduction to Function Spaces on the Disk (Matematički Institut SANU, Belgrade, 2004).Google Scholar
Pavlović, M., ‘Lipschitz conditions on the modulus of a harmonic function’, Rev. Mat. Iberoam. 23 (2007), 831–1845.Google Scholar
Pavlović, M., ‘A Schwarz lemma for the modulus of a vector-valued analytic function’, Proc. Amer. Math. Soc. 139 (2011), 969–973.Google Scholar
Xinzhong, H., ‘Estimates on Bloch constants for planar harmonic mappings’, J. Math. Anal. Appl. 337 (2008), 880–887.Google Scholar