Abstract
In this paper, we will use the Schwarz lemma at the boundary to character the distortion theorems of determinant at the extreme points and distortion theorems of matrix on the complex tangent space at the extreme points for normalized locally biholomorphic quasi-convex mappings in the unit ball B n respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Duren, P. L.: Univalent Functions, Grundlehren der mathematischen Wtssenschaften 259, Springer-Verlag, New York, 1983
Gong, S.: The Bieberbach Conjecture, International Press Company, Providence, 1999
Roper, K. A., Suffridge T. J.: Convexity properties of holomorphic mappings in Cn. Trans. Amer. Math. Soc., 351 (5), 1803–1833 (1999)
Liu, T. S., Liu, H.: Quasi-convex mappings on bounded convex circular domains. Acta Math. Sinica, Chinese Ser., 44 (2), 287–292 (2001)
Barnard, R. W., Fitzgerald, C. H., Gong, S.: A distortion theorem for biholomorphic mappings in C2. Trans. Amer. Math. Soc., 344 (2), 907–924 (1994)
Liu, T. S., Zhang, W. J.: A distortion theorem for biholomorphic convex mappings in Cn. Chin. J. Contemporary Math., 20 (3), 421–430 (1999)
Gong, S., Wang, S. K., Yu, Q. H.: Biholomorphic convex mappings of ball in Cn. Pac. J. Math., 161 (2), 287–306 (1993)
Pfaltzgraff, J. A., Suffridge, T. J.: Norm order and geometric properties of holomorphic mappings in Cn. J. Anal. Math., 82, 285–313 (2000)
Gong, S., Yu, Q. H.: The distortion theorems of linear invariant family on the unit ball. Asian J. Math., 4 (4), 795–816 (2000)
Chu, C. H., Hamada, H., Honda, T., et al.: Distortion theorems for convex mappings on homogeneous balls. J. Math. Anal. Appl., 369, 437–442 (2010)
Liu, T. S., Lu, J., Wang, J. F.: Distortion theorems of quasi-convex mappings in several complex variables. Chin. Ann. Math., 32A(5), 607–612 (2011)
Zhang, W. J., Liu, T. S.: On growth and covering theorems of quasi-convex mappings in the unit ball of a complex Banach space. Sci. China, Ser. A, 45 (12), 1538–1547 (2002)
Liu, T. S., Wang, J. F., Tang, X. M.: The Schwarz lemma at the boundary of the unit ball in Cn and its application. J. Geom. Anal., 25, 1890–1914 (2015)
Acknowledgements
We thank the referees for their time and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (Grant Nos. 11101139, 11031008, 11301136, 11571089, 11401159 and 11501198), the Science Foundation of Hebei Province (Grant No. A2014205069), the Key Scientific Research Projects in Universities of He’nan Province (Grant No. 16B110010), the Doctoral Foundation of Pingdingshan University (Grant No. PXY-BSQD-2015005), the Doctoral Foundation of Hebei Normal University (Grant No. L2015B04), the Foster Foundation of Pingdingshan University (Grant No. PXYPYJJ2016007)
Rights and permissions
About this article
Cite this article
Zhang, X.F., Liu, T.S. & Xie, Y.H. Distortion theorems for normalized biholomorphic quasi-convex mappings. Acta. Math. Sin.-English Ser. 33, 1242–1248 (2017). https://doi.org/10.1007/s10114-017-6220-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-017-6220-5