Sharp distortion theorems for a subclass of biholomorphic mappings which have a parametric representation in several complex variables. (English) Zbl 1350.32009
Summary: In this paper, the sharp distortion theorems of the Fréchet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in \(\mathbb{C}^n\) are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in \(\mathbb{C}^n\) are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in \(\mathbb{C}^n\) are got as well. Thus, some known results in prior literatures are generalized.
MSC:
| 32A30 | Other generalizations of function theory of one complex variable |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
| 46G20 | Infinite-dimensional holomorphy |
References:
| [1] | Poreda, T., On the univalent holomorphic maps of the unit polydisc of Cn which have the parametric representation, I—the geometric properties, Ann. Univ. Mariae Curie Sklodowsda Sect. A, 41, 1987, 105-113. · Zbl 0698.32004 |
| [2] | Kohr, G., Using the method of Löwner chains to introduce some subcalsses of biholomorphic mappings in Cn, Rev. Roum. Math. Pures Appl., 46, 2001, 743-760. · Zbl 1036.32014 |
| [3] | Graham, I., Hamada, H. and Kohr, G., Parametric representation of univalent mappings in several complex variables, Canadian J. Math., 54(2), 2002, 324-351. · Zbl 1004.32007 · doi:10.4153/CJM-2002-011-2 |
| [4] | Kubicka, E. and Poreda, T., On the parametric representation of starlike maps of the unit ball in Cn into Cn, Demonstration Math., 21, 1988, 345-355. · Zbl 0674.32014 |
| [5] | Kohr, G. and Liczberski, P., Univalent Mappings of Several Complex Variables, Cluj University Press, Cluj-Napoca, Romania, 1998. · Zbl 0926.32008 |
| [6] | Hamada, H., Honda, T. and Kohr, G., Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation, J. Math. Anal. Appl., 317(1), 2006, 302-319. · Zbl 1092.32011 · doi:10.1016/j.jmaa.2005.08.002 |
| [7] | Hamada, H. and Honda, T., Sharp growth theorems and coefficient bounds for starlike mappings in several complex variables, Chin. Ann. Math. Ser. B, 29(4), 2008, 353-368. · Zbl 1165.32006 · doi:10.1007/s11401-007-0339-0 |
| [8] | Xu, Q. H. and Liu, T. S., Sharp growth and distortion theorems for a subclass of biholomorphic mappings, Computer Math. Appl., 59(12), 2010, 3778-3784. · Zbl 1198.30015 · doi:10.1016/j.camwa.2010.04.012 |
| [9] | Xu, Q. H., Liu, T. S. and Xu, H. M., Growth and distortion theorems for a subclass of holomorphic mappings in several complex variables, Chin. Ann. Math. Ser. A, 35(5), 2014, 565-574 (in Chinese). · Zbl 1324.32009 |
| [10] | Barnard, R. W., FitzGerald C. H. and Gong S., A distortion theorem of biholomorphic convex mappings in C2, Trans. Amer. Math. Soc., 344(2), 1994, 907-924. · Zbl 0814.32004 |
| [11] | Liu, T. S. and Zhang, W. J., A distortion theorem of biholomorphic convex mappings in Cn, Chin. Ann. Math. Ser. A, 20(4), 1999, 505-512 (in Chinese). · Zbl 0978.32015 |
| [12] | Gong, S., Wang, S. K. and Yu, Q. H., Biholomorphic convex mappings of ball in Cn, Pacif. J. Math., 161(2), 1993, 287-306. · Zbl 0788.32017 · doi:10.2140/pjm.1993.161.287 |
| [13] | Gong, S. and Liu, T. S., Distortion theorems for biholomorphic convex mappings on bounded convex circular domains, Chin. Ann. Math. Ser. B, 20(3), 1999, 297-304. · Zbl 0942.32017 · doi:10.1142/S0252959999000321 |
| [14] | Liu, T. S. and Zhang, W. J., A distortion theorem of biholomorphic convex mappings in a Banach space, Acta Math. Sin., 46(6), 2003, 1041-1046 (in Chinese). · Zbl 1042.32005 |
| [15] | Zhu, Y. C. and Liu, M. S., Distortion theorems for biholomorphic convex mappings in Banach spaces, Complex Variables, 50(1), 2005, 57-68. · Zbl 1083.32014 · doi:10.1080/02781070412331329412 |
| [16] | Chu, C. H., Hamada, H., Honda T. and Kohr G., Distortion theorems for convex mappings on homogeneous balls, J. Math. Anal. Appl., 369(2), 2010, 437-442. · Zbl 1208.32015 · doi:10.1016/j.jmaa.2010.03.014 |
| [17] | Hamada, H. and Kohr, G., Growth and distortion results for convex mappings in infinite dimensional spaces, Complex Variables, 47(4), 2002, 291-301. · Zbl 1034.46043 · doi:10.1080/02781070290013866 |
| [18] | Liu, T. S., Wang, J. F. and Lu, J., Distortion theorems for starlike mappings in several complex variables, Taiwanese J. Math., 15(6), 2011, 2601-2608. · Zbl 1246.32002 |
| [19] | Liu, X. S. and Liu, T. S., On the sharp distortion theorems for a subclass of starlike mappings in several complex variables, Taiwanese J. Math., 19(2), 2015, 363-379. · Zbl 1357.32014 |
| [20] | Hamada, H., Kohr, G. and Liczberski, P., Starlike mappings of order a on the unit ball in complex Banach spaces, Glas. Mat. Ser. III, 36(1), 2001, 39-48. · Zbl 0985.32003 |
| [21] | Xu, Q. H. and Liu, T. S., On coefficient estimates for a class of holomorphic mappings, Sci. China Ser. A-Math.52(4), 2009, 677-686. · Zbl 1195.32011 · doi:10.1007/s11425-008-0132-x |
| [22] | Kohr, G. and Liczberski, P., On strongly starlikeness of order a in several complex variables, Glas. Mat., 33(53), 1998, 185-198. · Zbl 0926.32008 |
| [23] | Feng, S. X. and Lu, K. P., The growth theorem for almost starlike mappings of order a on bounded starlike circular domains, Chin. Quart. J. Math., 15(2), 2000, 50-56. · Zbl 0982.32002 |
| [24] | Chuaqui, M., Applications of subordination chains to starlike mappings in Cn, Pacif. J. Math., 168(1), 1995, 33-48. · Zbl 0822.32001 · doi:10.2140/pjm.1995.168.33 |
| [25] | Honda, T., The growth theorem for k-fold symmetric convex mappings, Bull. London Math. Soc., 34(6), 2002, 717-724. · Zbl 1029.32007 · doi:10.1112/S0024609302001388 |
| [26] | Lin, Y. Y. and Hong, Y., Some properties of holomorphic maps in Banach spaces, Acta Math. Sin., 38(2), 1995, 234-241 (in Chinese). · Zbl 0828.46045 |
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