The Coefficient Inequalities for a Class of Holomorphic Mappings in Several Complex Variables

Abstract

The authors establish the coefficient inequalities for a class of holomorphic mappings on the unit ball in a complex Banach space or on the unit polydisk in ℂ ⁿ, which are natural extensions to higher dimensions of some Fekete and Szegö inequalities for subclasses of the normalized univalent functions in the unit disk.

Keywords

Coefficient inequality / Fekete-Szegö problem / Quasi-convex mappings

References

[1] Fekete M, Szegö G. Eine Bemerkunguber ungerade schlichte Funktionen. J. Lond. Math. Soc., 1933, 8: 85-89

[2] Graham I, Hamada H, Honda T Growth, distortion and coefficient bounds for Carathéodory families in ℂ ⁿ and complex Banach spaces. J. Math. Anal. Appl., 2014, 416: 449-469

[3] Graham I, Hamada H, Kohr G. Parametric representation of univalent mappings in several complex variables. Canadian J. Math., 2002, 54: 324-351

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

[5] Graham I, Kohr G, Kohr M. Loewner chains and parametric representation in several complex variables. J. Math. Anal. Appl., 2003, 281: 425-438

[6] Hamada H, Honda T. Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables. Chin. Ann. Math. Ser. B, 2008, 29: 353-368

[7] 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

[8] Keogh F R, Merkes E P. A coefficient inequality for certain classes of analytic functions. Proc. Amer. Math. Soc., 1969, 20: 8-12

[9] Kohr G. On some best bounds for coefficients of several subclasses of biholomorphic mappings in ℂⁿ. Complex Variables, 1998, 36: 261-284

[10] Liu X S, Liu M S. Quasi-convex mappings of order α on the unit polydisc in ℂⁿ. Rocky Mountain Journal of Mathematics, 2010, 40(5): 1619-1643

[11] Liu X S, Liu T S. The sharp estimates of all homogeneous expansions for a class of quasi-convex mappings on the unit polydisk in ℂⁿ. Chin. Ann. Math. Ser. B, 2011, 32: 241-252

[12] Robertson M S. On the theory of univalent functions. Ann. Math., 1936, 37: 374-408

[13] Roper K, Suffridge T J. Convexity properties of holomorphic mappings in ℂⁿ. Trans. Amer. Math. Soc., 1999, 351: 1803-1833

[14] Xu Q H, Fang F, Liu T S. On the Fekete and Szego problem for starlike mappings of order α. Acta Math. Sin. (Engl. Ser.), 2017, 33: 1-11

[15] Xu Q H, Liu T S. On coefficient estimates for a class of holomorphic mappings. Sci China Math., 2009, 52: 677-686

[16] Xu Q H, Liu T S. On the Fekete and Szego problem for the class of starlike mappings in several complex variables. Abstr. Appl. Anal., 2014, 2014: 807026

[17] Xu Q H, Liu T S, Liu X S. The sharp estimates of homogeneous expansions for the generalized class of close-to-quasi-convex mappings. J. Math. Anal. Appl., 2012, 389: 781-791

[18] Xu Q H, Yang T, Liu T S, Xu H M. Fekete and Szego problem for a subclass of quasi-convex mappings in several complex variables. Front. Math. China, 2015, 10: 1461-1472