Abstract
In this paper, we first establish three new Landau-type theorems of polyharmonic mappings, which extend the related results of biharmonic mappings of earlier authors. Then three new Landau-type theorems of log-p-harmonic mappings are also provided.
Similar content being viewed by others
References
Chen, H.-H., Gauthier, P.M., Hengartner, W.: Bloch constants for planar harmonic mappings. Proc. Am. Math. Soc. 128, 3231–3240 (2000)
Chen, S.H., Ponnusamy, S., Wang, X.: Bloch constant and Landau’s theorem for planar p-harmonic mappings. J. Math. Anal. Appl. 373, 102–110 (2011)
Chen, S.H., Ponnusamy, S., Wang, X.: Coefficient estimates and Landau–Bloch’s constant for harmonic mappings. Bull. Malays. Math. Sci. Soc. (2) 34(2), 255–265 (2011)
Colonna, F.: The Bloch constant of bounded harmonic mappings. Indiana Univ. Math. J. 38(4), 829–840 (1989)
Dorff, M., Nowark, M.: Landau’s theorem for planar harmonic mappings. Comput. Methods Funct. Theory 4, 151–158 (2004)
Grigoryan, A.: Landau and Bloch theorems for planar harmonic mappings. Complex Var. Elliptic Equ. 51, 81–87 (2006)
Landau, E.: Der Picard–Schottysche Satz und die Blochsche Konstanten, pp. 467–474. Sitzungsber Press, Berlin (1929)
Li, D.Z., Chen, X.D.: Bloch constant of harmonic mappings. J. Huaqiao Univ. (Nat. Sci.) 33(1), 103–106 (2012)
Li, P., Ponnusamy, S., Wang, X.: Some properties of planar \(p\)-harmonic and log-\(p\)-harmonic mappings. Bull. Malays. Math. Sci. Soc. (2) 36(3), 595–609 (2013)
Li, P., Wang, X.: Landau’s theorem for log-p-harmonic mappings. Appl. Math. Comput. 218, 4806–4812 (2012)
Liu, M.S.: Landau’s theorems for biharmonic mappings. Complex Var. Elliptic Equ. 53, 843–855 (2008)
Liu, M.S.: Estimates on Bloch constants for planar harmonic mappings. Sci. China Ser. A Math. 52(1), 87–93 (2009)
Liu, M.S.: Landau’s theorems for planar harmonic mappings. Comput. Math. Appl. 57(7), 1142–1146 (2009)
Liu, M.S., Liu, Zhen-Xing: Landau-type theorems for p-harmonic mappings or log-p-harmonic mappings. Appl. Anal. 51(1), 81–87 (2014)
Liu, M.S., Xie, L., Yang, L.M.: Landau’s theorems for biharmonic mappings(II). Math. Methods Appl. Sci. 40(7), 2582–2595 (2017)
Mao, Zh, Ponnusamy, S., Wang, X.: Schwarzian derivative and Landau’s theorem for logharmonic mappings. Complex Var. Elliptic Equ. 58(8), 1093–1107 (2013)
Xia, X.Q., Huang, X.Z.: Estimates on Bloch constants for planar bounded harmonic mappings (in chinese). Chin. Ann. Math. 31A(6), 769–776 (2010)
Acknowledgements
The authors are grateful to the anonymous referees for making many valuable suggestions that improved the quality and the readability of this paper. The research was financially supported by Guangdong Natural Science Foundation (Grant No.2014A030313422).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Dan Volok.
Rights and permissions
About this article
Cite this article
Bai, XX., Liu, MS. Landau-Type Theorems of Polyharmonic Mappings and log-p-Harmonic Mappings. Complex Anal. Oper. Theory 13, 321–340 (2019). https://doi.org/10.1007/s11785-018-0784-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-018-0784-7