Frontiers of Mathematics in China >
Growth and distortion theorems on subclasses of quasi-convex mappings in several complex variables
Received date: 29 May 2011
Accepted date: 15 Aug 2011
Published date: 01 Oct 2011
Copyright
In this paper, the refining growth and covering theorems for f are established, where f is a quasi-convex mapping of order α and x = 0 is a zero of order k + 1 of f(x) - x. As an application, we obtain the upper and lower bounds on the distortion theorem of f(x) defined on the unit polydisc of . The upper bound of the distortion theorem for f(x) defined on the unit ball of a complex Banach space is also given. Our results extend the growth and distortion theorems for convex functions of one complex variable to quasi-convexmappings of several complex variables.
Jianfei WANG , Taishun LIU , Jin LU . Growth and distortion theorems on subclasses of quasi-convex mappings in several complex variables[J]. Frontiers of Mathematics in China, 2011 , 6(5) : 931 -944 . DOI: 10.1007/s11464-011-0158-1
| 1 |
Barnard RW, Fitzgerald C H, Gong S. Distortion theorem for biholomorphic mappings in
|
| 2 |
Chu C H, Hamada H, Honda T, Kohr G. Distortion theorems for convex mappings on homogeneous balls. J Math Anal Appl, 2010, 369: 437-442
|
| 3 |
Duren P L. Univalent Function. Berlin: Springer-Verlag, 1983
|
| 4 |
Gong S. Convex and Starlike Mappings in Several Complex Variables. Beijing: Science Press/Kluwer Academic Publishers, 1998
|
| 5 |
Gong S, Liu T S. Distortion theorems for biholomorphic convex mappings on bounded circular convex domains. Chin Ann Math, 1999, 20: 297-304
|
| 6 |
Gong S, Wang S K, Yu Q H. Biholomorphic convex mappings of ball in
|
| 7 |
Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker, 2003
|
| 8 |
Graham I, Varolin D. Bloch constants in one and several variables. Pacific J Math, 1996, 174: 347-357
|
| 9 |
Honda T. The growth theorem for k-fold symmetric convex mappings. Bull London Math Soc, 2002, 34: 717-724
|
| 10 |
Kato T. Nonlinear semigroups and evolution equations, J Math Soc Japan, 1967, 19: 508-520
|
| 11 |
Kikuchi K. Starlike and convex maps in several complex variables. Pacific J Math, 1973, 44: 569-580
|
| 12 |
Liu T S, Liu H. Quasi-convex Mappings on bounded convex circular domains. Acta Math Sinica, 2001, 44: 287-292 (in Chinese)
|
| 13 |
Liu T S, Liu X S. A refinement about estimation of expansion coefficients for normalized biholomorphic mappings. Sci China Math, 2005, 48: 865-879
|
| 14 |
Liu T S, Ren G B. Growth theorem of convex mappings on bounded convex circular domains. Sci China Math, 1998, 41: 123-130
|
| 15 |
Liu T S, Xu Q H. On quasi-convex mappings of order α in the unit ball of a complex Banach space. Sci China Math, 2006, 49: 1451-1457
|
| 16 |
Liu T S, Zhang W J. A distortion theorem of biholomorphic convex mappings in a Banach space. Acta Math Sinica, 2003, 46: 1041-1046 (in Chinese)
|
| 17 |
Roper K A, Suffridge T J. Convexity properties of holomorphic mappings in
|
| 18 |
Suffridge T J. The principal of subordination applied to functions of several variables. Pacific J Math, 1970, 33: 241-248
|
| 19 |
Suffridge T J. Starlike and convex maps in Banach spaces. Pacific J Math, 1973, 46: 575-589
|
| 20 |
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 Math, 2002, 45: 1535-1547
|
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