Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables

Abstract

Let B be the unit ball in a complex Banach space. Let S _k+1 ^*(B) be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k+1 of f(z)-z. The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S _k+1 ^*(B).

Keywords

Sharp coefficient bound / Sharp covering theorem / Sharp growth theorem / Starlike mapping / Zero of order k

References

[1] Ali R. M.. Starlikeness associated with parabolic regions. Int. J. Math. Math. Sci., 2005, 2005: 561-570

[2] Barnard R. W., FitzGerald C. H., Gong S.. The growth and $\frac{1}{4}$-theorems for starlike functions in ℂⁿ. Pacific J. Math., 1991, 150: 13-22

[3] Cartan H.. Montel P.. Sur la possibilité d’étendre aux fonctions de plusieurs variables complexes, La théorie des fonctions univalents. Lecons sur les Fonctions Univalents ou Multivalentes, 1933, Pairs: Gauthier-Villars 129-155

[4] Chuaqui M.. Applications of subordination chains to starlike mappings in ℂⁿ. Pacific J. Math., 1995, 168: 33-48

[5] Feng S., Lu K.. The growth theorem for almost starlike mappings of order α on bounded starlike circular domains. Chin. Quart. J. Math., 2000, 15: 50-56

[6] FitzGerald C. H., Thomas C.. Some bounds on convex mappings in several complex variables. Pacific J. Math., 1994, 165: 295-320

[7] Gong S.. The Bieberbach Conjecture. Amer. Math. Soc., 1999, Cambridge, MA: Intern. Press

[8] Graham I., Hamada H., Kohr G.. Parametric representation of univalent mappings in several complex variables. Canad. J. Math., 2002, 54(2): 324-351

[9] Graham I., Kohr G.. Geometric Function Theory in One and Higher Dimensions, 2003, New York: Marcel Dekker Inc.

[10] Gurganus K.. Φ-like holomorphic functions in ℂⁿ and Banach spaces. Trans. Amer. Math. Soc., 1975, 205: 389-406

[11] Hamada H.. Starlike mappings on bounded balanced domains with C ¹-plurisubharmonic defining functions. Pacific J. Math., 2000, 194: 359-371

[12] Hamada H., Honda T., Kohr G.. Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation. J. Math. Anal. Appl., 2006, 317: 302-319

[13] Hamada H., Honda T., Kohr G.. Parabolic starlike mappings in several complex variables. Manuscripta Math., 2007, 123: 301-324

[14] Hamada H., Kohr G.. Subordination chains and the growth theorem of spirallike mappings. Mathematica (Cluj), 2000, 42(65): 153-161

[15] Hamada H., Kohr G.. Growth and distortion results for convex mappings in infinite dimensional spaces. Complex Var. Theory Appl., 2002, 47: 291-301

[16] Hamada H., Kohr G.. Φ-like and convex mappings in infinite dimensional spaces. Rev. Roumaine Math. Pures Appl., 2002, 47: 315-328

[17] Hamada H., Kohr G., Liczberski P.. Starlike mappings of order α on the unit ball in complex Banach spaces. Glas. Mat. Ser. III, 2001, 36(56): 39-48

[18] Honda T.. The growth theorem for k-fold symmetric convex mappings. Bull. London Math. Soc., 2002, 34: 717-724

[19] Kohr G.. Certain sufficient conditions of almost starlikeness of order ½ on the unit ball in ℂⁿ. Zeszyty Nauk. Politech. Rzeszowskiej Mat., 1996, 19: 13-25

[20] Kohr G.. On starlikeness and strongly starlikeness of order α in ℂⁿ. Mathematica (Cluj), 1998, 40(63): 95-109

[21] Kohr G.. On some best bounds for coefficients of subclasses of univalent holomorphic mappings in ℂⁿ. Complex Var. Theory Appl., 1998, 36: 261-284

[22] Kohr G.. Using the method of Löwner chains to introduce some subclasses of univalent holomorphic mappings in ℂⁿ. Rev. Roumaine Math. Pures Appl., 2001, 46: 743-760

[23] Liu H., Li X.. The growth theorem for strongly starlike mappings of order α on bounded starlike circular domains. Chin. Quart. J. Math., 2000, 15: 28-33

[24] Liu T., Liu X.. On the precise growth, covering and distortion theorem for normalized biholomorphic mappings. J. Math. Anal. Appl., 2004, 295: 404-417

[25] Liu T., Liu X.. A refinement about estimation of expansion coefficients for normalized biholomorphic mappings. Sci. in China, Ser. A, 2005, 48: 865-879

[26] Lloyd N.. Degree Theory, 1978, Cambridge: Cambridge University Press

[27] Ma W., Minda D.. A unified treatment of some special classes of univalent functions. Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 1994, Cambridge, MA: Internat. Press 157-169

[28] Pommerenke C.. Univalent Functions, 1975, Göttingen: Vandenhoeck & Ruprecht

[29] Poreda T.. On the univalent holomorphic maps of the unit polydisc in ℂⁿ which have the parametric representation, I-the geometrical properties. Ann. Univ. Mariae Curie Sklodowska, Sect. A, 1987, 41: 105-113

[30] Rogosinski W.. On the coefficients of subordinate functions. Proc. London Math. Soc. (2), 1943, 48: 48-82

[31] Rønning F.. On starlike functions associated with parabolic regions. Ann. Univ. Mariae Curie-Sk lodowska Sect. A, 1991, 45: 117-122

[32] Rønning F.. Uniformly convex functions and a corresponding class of starlike functions. Proc. Amer. Math. Soc., 1993, 118: 189-196

[33] Suffridge T. J.. Starlike and convex maps in Banach spaces. Pacific J. Math., 1973, 46: 474-489

[34] Suffridge T. J.. Biholomorphic mappings of the ball onto convex domains. Abstract Amer. Math. Soc., 1990, 11(66): 46

[35] Zhang W. J., Dong D. Z.. A growth theorem and a ¼-theorem for starlike mappings in Banach spaces (in Chinese). Chin. Ann. Math., 1992, 13A(4): 417-423