Abstract
A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ℂn, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.
Similar content being viewed by others
References
Duren, P. L., Univalent Functions, Berlin: Springer-Verlag, 1983.
Kikuchi, K., Starlike and convex mappings in several complex variables, Pacific J. of Math., 1973, 44(3): 569–580.
Suffridge, T. J., Starlike and convex maps in Banach spaces, Pacific J. of Math., 1973, 46(3): 575–589.
Gong Sheng, Wang Shikun, Yu Qihuang, Biholomorphic convex mappings of ball in ℂn, Pacific Jour. of Math., 1993, 161(2): 287–306.
Liu Taishun, The growth theorem and covering theorems for biholomorphic mappings on classical domains, University of Science and Technology of China, Doctor Thesis, 1989.
Liu Taishun, Ren Guangbin. The Growth theorem of convex mapping on bounded convex circular domains, Science in China(Series A), 1998, 41(2): 123–130.
Thomas, C. R., Extensions of classical results in one complex variables to several complex variables, University of California, San Diego, Doctor Thesis, 1991.
Roper, K. A., Suffridge, T. J., Convexity properties of holomorphic mappings in ℂn, Tran. Amer. Soc., 1999, 351(5): 1803–1833.
Liu Taishun, Liu Hao, Quasi-convex Mappings on bounded convex circular domains, Acta Math. Sinica, 2001, 44(2): 287–292.
Gong Sheng, Convex and Starlike Mappings in Several Complex Variables, Science Press/Kluwer Academic Publishers, 1998.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wenjun, Z., Taishun, L. On growth and covering theorems of quasi-convex mappings in the unit ball of a complex banach space. Sci. China Ser. A-Math. 45, 1538–1547 (2002). https://doi.org/10.1360/02ys9165
Received:
Issue Date:
DOI: https://doi.org/10.1360/02ys9165