Transactions of the American Mathematical Society

Author(s): Kevin A. Roper; Ted J. Suffridge
Journal: Trans. Amer. Math. Soc. 351 (1999), 1803-1833.
MSC (1991): Primary 32H99; Secondary 30C45
Posted: January 26, 1999
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract: Not many convex mappings on the unit ball in ${\mathbb C}^n$ for

are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the ``Quasi-Convex Mappings, Types A and B'' seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.

1.: C. H. Fitzgerald & C. R. Thomas, Some Bounds on Convex Mappings in Several Complex Variables, Pacific Journal of Math. 2(1994), 295-320. MR 95k:32021
2.: S. Gong, S. Wang & Q. Yu, Biholomorphic Convex Mappings of Balls in ${\bf C^n}$ , Pacific J. Math. 161(1993), 287-306. MR 94i:32029
3.: K. R. Gurganus. $\Phi$ -like holomorphic functions in ${\bf C^n}$ and Banach spaces. Trans. Amer. Math. Soc. 205(1975), 389-406. MR 51:10670
4.: L. A. Harris, Schwarz's Lemma in normed linear spaces, Proc. Natl. Acad. Sci. 62 (1969), 1014-1017. MR 43:936
5.: L. Hörmander, On a Theorem of Grace, Math. Scandia. 2(1954), 55-64. MR 16:27b
6.: T. Matsuno, Starlike theorems and convex-like theorems in the complex vector space, Sci. Rep. Tokyo, Kyoiku Daigaku, Sect. A, 5(1955), 88-95. MR 18:329d
7.: Z. Nehari. A Property of Convex Conformal Maps, Journal D'Analyse Mathématique. 30(1976), 390-393. MR 55:12901
8.: -, Conformal Mapping, McGraw-Hill, 1952 (Now Dover (1975)). MR 13:640h; MR 51:13206
9.: M. S. Robertson, Applications of the subordination principle to univalent functions, Pacific Journal of Math. 11 (1961), 315-324. MR 23:A1787
10.: K. A. Roper & T. J. Suffridge. Convex Mappings on the Unit Ball of ${\mathbb C}^n$ , J. Anal. Math. 65 (1995), 333-347. MR 96m:32023
11.: T. J. Suffridge. The principle of subordination applied to functions of several variables. Pacific Journal of Math. 33(1970), 241-248. MR 41:5660
12.: -, Some Remarks on Convex Maps of the Unit Disk, Duke Mathematical Journal 37(1970), 775-777. MR 42:4722
13.: -, Starlike and convex maps in Banach spaces. Pacific Journal of Math. 46(1973), 575-589. MR 51:11110
14.: -, Starlikeness, Convexity and Other Geometric Properties of Holomorphic Maps in Higher Dimensions, Lecture Notes in Mathematics, vol 599, Springer-Verlag(1976), 146-159. MR 56:8894

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 32H99, 30C45

Kevin A. Roper
Affiliation: Department of Mathematics, Munro College, P.O., St. Elizabeth, Jamaica

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS