Convexity properties of holomorphic mappings in $\mathbb {C}^n$
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- by Kevin A. Roper and Ted J. Suffridge
- Trans. Amer. Math. Soc. 351 (1999), 1803-1833
- DOI: https://doi.org/10.1090/S0002-9947-99-02219-9
- Published electronically: January 26, 1999
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Abstract:
Not many convex mappings on the unit ball in ${\mathbb C}^n$ for $n>1$ 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.References
- Carl H. FitzGerald and Carolyn R. Thomas, Some bounds on convex mappings in several complex variables, Pacific J. Math. 165 (1994), no. 2, 295–320. MR 1300835, DOI 10.2140/pjm.1994.165.295
- Sheng Gong, Shi Kun Wang, and Qi Huang Yu, Biholomorphic convex mappings of ball in $\textbf {C}^n$, Pacific J. Math. 161 (1993), no. 2, 287–306. MR 1242200, DOI 10.2140/pjm.1993.161.287
- Kenneth R. Gurganus, $\Phi$-like holomorphic functions in $\textbf {C}^{n}$ and Banach spaces, Trans. Amer. Math. Soc. 205 (1975), 389–406. MR 374470, DOI 10.1090/S0002-9947-1975-0374470-1
- Lawrence A. Harris, Schwarz’s lemma in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1014–1017. MR 275179, DOI 10.1073/pnas.62.4.1014
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
- Zeev Nehari, A property of convex conformal maps, J. Analyse Math. 30 (1976), 390–393. MR 440020, DOI 10.1007/BF02786725
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- M. S. Robertson, Applications of the subordination principle to univalent functions, Pacific J. Math. 11 (1961), 315–324. MR 124475, DOI 10.2140/pjm.1961.11.315
- Kevin A. Roper and Ted J. Suffridge, Convex mappings on the unit ball of $\textbf {C}^n$, J. Anal. Math. 65 (1995), 333–347. MR 1335379, DOI 10.1007/BF02788776
- T. J. Suffridge, The principle of subordination applied to functions of several variables, Pacific J. Math. 33 (1970), 241–248. MR 261040, DOI 10.2140/pjm.1970.33.241
- T. J. Suffridge, Some remarks on convex maps of the unit disk, Duke Math. J. 37 (1970), 775–777. MR 269827, DOI 10.1215/S0012-7094-70-03792-0
- T. J. Suffridge, Starlike and convex maps in Banach spaces, Pacific J. Math. 46 (1973), 575–589. MR 374914, DOI 10.2140/pjm.1973.46.575
- T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976) Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977, pp. 146–159. MR 0450601
Bibliographic Information
- Kevin A. Roper
- Affiliation: Department of Mathematics, Munro College, P.O., St. Elizabeth, Jamaica
- Ted J. Suffridge
- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- Email: ted@ms.uky.edu
- Received by editor(s): July 10, 1995
- Received by editor(s) in revised form: August 11, 1997
- Published electronically: January 26, 1999
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 1803-1833
- MSC (1991): Primary 32H99; Secondary 30C45
- DOI: https://doi.org/10.1090/S0002-9947-99-02219-9
- MathSciNet review: 1475692