We mainly discuss the invariance of some subclasses of biholomorphic mappings under the generalized Roper-Suffridge operators on Bergman-Hartogs domains which are based on the unit ball Bn. Using the geometric properties and the distortion results of subclasses of biholomorphic mappings, we obtain the geometric characters of almost spirallike mappings of type β and order α, SΩ*(β, A, B), strong and almost spirallike mappings of type β and order α maintained under the generalized Roper-Suffridge operators on Bergman-Hartogs domains. Sequentially, we conclude that the generalized operators and the known operators preserve the same properties under some conditions. The conclusions generalize some known results.
Yanyan CUI
,
Hao LIU
. THE INVARIANCE OF SUBCLASSES OF BIHOLOMORPHIC MAPPINGS ON BERGMAN-HARTOGS DOMAINS[J]. Acta mathematica scientia, Series B, 2019
, 39(4)
: 1103
-1120
.
DOI: 10.1007/s10473-019-0414-0
[1] Roper K A, Suffridge T J. Convex mappings on the unit ball of Cn. J Anal Math, 1995, 65:333-347
[2] Graham I, Kohr G. Univalent mappings associated with the Roper-Suffridge extension operator. J Analyse Math, 2000, 81:331-342
[3] Liu T S, Gong S. Family of ε starlike mappings (I). Chin Ann Math, 2002, 23A(3):273-282
[4] Liu X S, Liu T S. On the generalized Roper-Suffridge extension operator for spirallike mappings of type β and order α. Chin Ann Math, 2006, 27A(6):789-798
[5] Feng S X, Zhang X F, Chen H Y. The generalized Roper-Suffridge extension operator. Chin Ann Math, 2011, 54A(3):467-482
[6] Wang J, Liu T. The Roper-Suffridge extension operator and its applications to convex mappings in C2. Trans Amer Math Soc, 2018. DOI:https://doi.org/10.1090/tran/7221
[7] Wang J F, Liu T S. A modification of the Roper-Suffridge extension operator for some holomorphic mappings. Chin Ann Math, 2010, 31A(4):487-496
[8] Muir J R. A modification of the Roper-Suffridge extension operator. Comput Methods Funct Theory, 2005, 5(1):237-251
[9] Muir J R, Suffridge T J. Unbounded convex mappings of the ball in Cn. Trans Amer Math Soc, 2001, 129:3389-3393
[10] Kohr G. Loewner chains and a modification of the Roper-Suffridge extension operator. Mathematica, 2006, 71(1):41-48
[11] Muir J R. A class of Loewner chain preserving extension operators. J Math Anal Appl, 2008, 337(2):862-879
[12] Tang Y Y. Roper-Suffridge operators on Bergman-Hartogs domain[D]. Kaifeng:Henan University Master Thesis, 2016
[13] Zhu Y C, Liu M S. The generalized Roper-Suffridge extension operator on Reinhardt domain Dp. Taiwanese Jour of Math, 2010, 14(2):359-372
[14] Feng S X, Liu T S, Ren G B. The growth and covering theorems for several mappings on the unit ball in complex Banach spaces. Chin Ann Math, 2007, 28A(2):215-230
[15] Gao C L. The generalized Roper-Suffridge operators on Reinhardt domains[D]. Jinhua:Zhejiang Normal University Master Thesis, 2012
[16] Liu X S, Feng S X. A remark on the generalized Roper-Suffridge extension operator for spirallike mappings of type β and order α. Chin Quart J of Math, 2009, 24(2):310-316
[17] Cai R H, Liu X S. The third and fourth coefficient estimations for the subclasses of strongly spirallike functions. Journal of Zhanjiang Normal College, 2010, 31(6):38-43
[18] Liu T S, Ren G B. Growth theorem for starlike mappings on bounded starlike circular domains. Chin Ann of Math, 1998, 19B(4):401-408
[19] Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York:Marcel Dekker, 2003
[20] Zhang J, Lu J. Distortion theorems of almost spirallike mappings of type β with order α on the unit polydisc. Journal of Huzhou Teachers College, 2011, 33(2):46-50
[21] Duren P L. Univalent Functions. New York:Springer-Verlag, 1983
[22] Ahlfors L V. Complex Analysis. 3rd ed. New York:Mc Graw-Hill Book Co, 1979
[23] Hamada H, Kohr G. The growth theorem and quasiconformal extension of strongly spirallike mappings of type α. Complex Variables Theory and Application, 2001, 44(4):281-297
[24] Cui Y, Wang C, Liu H, et al. The distortion upper bounds for strongly spirallike mappings of type α. Mathematics in Practice and Theory, 2015, 45(13):258-262
[25] Wang C, Cui Y, Liu H. Property of the Modified Roper-Suffridge Extension Operators on Bn. Acta Mathematica Sinica, 2016, 59(6):729-744
[26] Wang C, Cui Y, Liu H. Properties of the modified Roper-Suffridge extension operators on Reinhardt domains. Acta Mathematica Scientia, 2016, 36B(6):1767-1779
[27] Wang C, Liu A, Cui Y, et al. Subclasses of spirallike mappings on Bn and the generalized Roper-Suffridge extension operators. Journal of Sichuan Normal University (Natural Science), 2016, 39(2):231-235
