# MAXIMUM PRINCIPLE, CONVERGENCE OF SEQUENCES AND ANGULAR LIMITS FOR HARMONIC BLOCH MAPPINGS

• Qiao, Jinjing ;
• Gao, Hongya
• Published : 2014.11.30
• 52 5

#### Abstract

In this paper, we investigate maximum principle, convergence of sequences and angular limits for harmonic Bloch mappings. First, we give the maximum principle of harmonic Bloch mappings, which is a generalization of the classical maximum principle for harmonic mappings. Second, by using the maximum principle of harmonic Bloch mappings, we investigate the convergence of sequences for harmonic Bloch mappings. Finally, we discuss the angular limits of harmonic Bloch mappings. We show that the asymptotic values and angular limits are identical for harmonic Bloch mappings, and we further prove a result that applies also if there is no asymptotic value. A sufficient condition for a harmonic Bloch mapping has a finite angular limit is also given.

#### Keywords

harmonic Bloch mapping;maximum principle;convergence;angular limit

#### References

1. J. J. Carmona, J. Cufi, and Ch. Pommerenke, On the angular limits of Bloch functions, Publ. Mat. 32 (1988), no. 2, 191-198. https://doi.org/10.5565/PUBLMAT_32288_07
2. J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37.
3. R. Attele, Bounded analytic functions and the little Bloch space, Internat. J. Math. Math. Sci. 13 (1990), no. 1, 193-198. https://doi.org/10.1155/S016117129000028X
4. F. G. Avkhadiev and K. J. Wirths, Schwarz-Pick Type Inequalities, Birkhauser, Basel, 2009.
5. SH. Chen, S. Ponnusamy, and X. Wang, Area integral means, Hardy and weighted Bergman spaces of planar harmonic mappings, Kodai Math. J. 36 (2013), no. 2, 313-324. https://doi.org/10.2996/kmj/1372337521
6. F. Colonna, The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), no. 4, 829-840. https://doi.org/10.1512/iumj.1989.38.38039
7. F. Colonna, Bloch and normal functions and their relation, Rend. Circ. Mat. Palermo (2) 38 (1989), no. 2, 161-180. https://doi.org/10.1007/BF02843992
8. P. Duren, Harmonic Mappings in the Plane, Cambridge Univ. Press, New York, 2004.
9. O. Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47-65. https://doi.org/10.1007/BF02392392
10. Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Gottingen, 1975.
11. A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47-76. https://doi.org/10.1215/S0012-7094-45-01206-3
12. A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1968.

#### Acknowledgement

Supported by : NSF