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On some classes of bounded univalent mappings in several complex variables. (English) Zbl 1208.32016
Let $$B^n$$ be the Euclidean ball in $$\mathbb{C}^n$$. In this paper, the authors study several properties of strongly starlike mappings of order $$\alpha\in (0,1)$$ and of bounded convex mappings on $$B^n$$. They show that $$K$$-quasiregular strongly starlike mappings of order $$\alpha$$ have continuous and univalent extensions to $$\overline{B^n}$$, and that bounded convex mappings on $$B^n$$ are strongly starlike of some order $$\alpha$$. They give a coefficient estimate for $$K$$-quasiregular strongly starlike mappings of order $$\alpha$$ in $$B^n$$, and give examples of such types of mappings.

##### MSC:
 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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##### References:
 [1] Bochner S.: Bloch’s theorem for real variables. Bull. Am. Math. Soc. 52, 715–719 (1946) · Zbl 0061.11204 [2] Brannan D.A., Kirwan W.E.: On some classes of bounded univalent functions. J. Lond. Math. Soc.(2) 1, 431–443 (1969) · Zbl 0177.33403 [3] Chen H., Gauthier P.M.: Bloch constants in several variables. Trans. Am. Math. Soc. 353, 1371–1386 (2001) · Zbl 0966.32002 [4] Chuaqui M.: Applications of subordination chains to starlike mappings in C n . Pac. J. Math. 168, 33–48 (1995) · Zbl 0822.32001 [5] FitzGerald C.H., Thomas C.R.: Some bounds on convex mappings in several complex variables. Pac. J. Math. 165, 295–320 (1994) · Zbl 0812.32003 [6] Gong S.: Convex and Starlike Mappings in Several Complex Variables. Kluwer Acadamic Publisher, Dordrecht (1998) · Zbl 0926.32007 [7] Graham I., Kohr G.: An extension theorem and subclasses of univalent mappings in several complex variables. Complex Var. Theory Appl. 47, 59–72 (2002) · Zbl 1026.32033 [8] Graham I., Kohr G.: Geometric Function Theory in One and Higher Dimensions. Marcel Dekker Inc., New York (2003) · Zbl 1042.30001 [9] Graham I., Hamada H., Kohr G.: Parametric representation of univalent mappings in several complex variables. Can. J. Math. 54, 324–351 (2002) · Zbl 1004.32007 [10] Graham I., Hamada H., Kohr G., Suffridge T.J.: Extension operators for locally univalent mappings. Mich. Math. J. 50, 37–55 (2002) · Zbl 1025.32017 [11] Graham I., Hamada H., Kohr G.: Radius problems for holomorphic mappings on the unit ball in C n . Math. Nachr. 279, 1474–1490 (2006) · Zbl 1116.32008 [12] Hallenbeck D.J., Ruscheweyh S.: Subordination by convex functions. Proc. Am. Math. Soc. 52, 191–195 (1975) · Zbl 0311.30010 [13] Hamada H., Honda T.: Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin. Ann. Math. B 29, 353–368 (2008) · Zbl 1165.32006 [14] Hamada H., Kohr G.: Simple criterions for strongly starlikeness and starlikeness of certain order. Math. Nachr. 254–255, 165–171 (2003) · Zbl 1029.32005 [15] Hamada H., Kohr G.: Quasiconformal extension of biholomorphic mappings in several complex variables. J. Anal. Math. 96, 269–282 (2005) · Zbl 1089.32009 [16] Hamada H., Honda T., Kohr G.: Parabolic starlike mappings in several complex variables. Manuscr. Math. 123, 301–324 (2007) · Zbl 1131.32009 [17] Hörmander L.: On a theorem of Grace. Math. Scand. 2, 55–64 (1954) · Zbl 0058.25502 [18] Kato T.: Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967) · Zbl 0163.38303 [19] Kohr G.: On starlikeness and strongly starlikeness of order alpha in C n . Mathematica (Cluj) 40(63), 95–109 (1998) · Zbl 1281.32013 [20] Kohr G., Liczberski P.: On strongly starlikeness of order alpha in several complex variables. Glas. Math. III 33(53), 185–198 (1998) · Zbl 0926.32008 [21] Krzyz J.: Distortion theorems for bounded convex functions. Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. 8, 625–627 (1960) · Zbl 0135.28902 [22] Liu, T.: The growth theorems, covering theorems and distortion theorems for biholomorphic mappings on classical domains. University of Science and Technology of China, Thesis (1989) [23] Liu H., Li X.: The growth theorem for strongly starlike mappings of order {$$\alpha$$} on bounded starlike circular domains. Chin. Q. J. Math. 15, 28–33 (2000) · Zbl 0983.32002 [24] Marden A., Rickman S.: Holomorphic mappings of bounded distortion. Proc. Am. Math. Soc. 46, 226–228 (1974) · Zbl 0301.32023 [25] Mercer P.R.: A general Hopf lemma and proper holomorphic mappings between convex domains in C n . Proc. Am. Math. Soc. 119, 573–578 (1993) · Zbl 0788.32021 [26] Pfaltzgraff J.A.: Subordination chains and quasiconformal extension of holomorphic maps in C n . Ann. Acad. Sci. Fenn. Ser. AI Math. 1, 13–25 (1975) · Zbl 0314.32001 [27] Pfaltzgraff J.A., Suffridge T.J.: An extension theorem and linear invariant families generated by starlike maps. Ann. Univ. Mariae Curie Skl. 53, 193–207 (1999) · Zbl 0996.32006 [28] Poletsky E.A.: Holomorphic quasiregular mappings. Proc. Am. Math. Soc. 95, 235–241 (1985) · Zbl 0582.32033 [29] Pommerenke C.: On starlike and convex functions. J. Lond. Math. Soc. 37, 209–224 (1962) · Zbl 0107.06501 [30] Roper K., Suffridge T.J.: Convex mappings of the unit ball in C n . J. Anal. Math. 65, 333–347 (1995) · Zbl 0846.32006 [31] Roper K., Suffridge T.J.: Convexity properties of holomorphic mappings in C n . Trans. Am. Math. Soc. 351, 1803–1833 (1999) · Zbl 0926.32012 [32] Suffridge T.J.: The principle of subordination applied to functions of several variables. Pac. J. Math. 33, 241–248 (1970) · Zbl 0196.09601 [33] Suffridge T.J.: Biholomorphic mappings of the ball onto convex domains. Abstr. pap. Present. Am. Math. Soc. 11(66), 46 (1990) · Zbl 0698.30011 [34] Suffridge, T.J.: Holomorphic mappings of domains in C N onto convex domains. In: FitzGerald, C.H., Gong, S. (eds.) Geometric Function Theory in Several Complex Variables, pp. 295–309. World Scientific Publishing, River Edge (2004) · Zbl 1071.32015 [35] Väisälä, J.: Lectures on n-Dimensional Quasiconformal Mappings. Lecture Notes in Mathematics, vol. 229. Springer, Berlin, New York (1971) · Zbl 0221.30031 [36] Wu H.: Normal families of holomorphic mappings. Acta Math. 119, 193–233 (1967) · Zbl 0158.33301
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