×

zbMATH — the first resource for mathematics

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.)
PDF BibTeX XML Cite
Full Text: DOI
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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.