Sharp distortion theorems for a subclass of close-to-convex mappings

doi:10.1007/s11464-013-0325-7

Front Math Chin

2013, Vol. 8

Issue (6) : 1425-1436 https://doi.org/10.1007/s11464-013-0325-7

RESEARCH ARTICLE

Sharp distortion theorems for a subclass of close-to-convex mappings

Qinghua XU¹(

), Taishun LIU², Xiaosong LIU³

1. College of Mathematics and Information Science, Jiangxi Normal University, Nanchang 330022, China; 2. Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China; 3. School of Mathematics and Computation Science, Zhanjiang Normal University, Zhanjiang 524048, China

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Abstract

We introduce the class of strongly close-to-convex mappings of order α in the unit ball of a complex Banach space, and then, we give the sharp distortion theorems for this class of mappings in the unit ball of a complex Hilbert space X or the unit polydisc in ?n. As an application, a sharp growth theorem for strongly close-to-convex mappings of order α is obtained.

Keywords Distortion theorem growth theorem strongly close-to-convex mappings of order α

Corresponding Author(s): XU Qinghua,Email:xuqh@mail.ustc.edu.cn

Issue Date: 01 December 2013

Cite this article:

Qinghua XU,Taishun LIU,Xiaosong LIU. Sharp distortion theorems for a subclass of close-to-convex mappings[J]. Front Math Chin, 2013, 8(6): 1425-1436.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-013-0325-7
https://academic.hep.com.cn/fmc/EN/Y2013/V8/I6/1425

1	Barnard R W, Fitzgerld C H, Gong S. A distortion theorem of biholomorphic convex mappings in ?2.Trans Amer Math Soc , 1994, 344: 907-924
2	Cartan H. Sur la possibilitéd’éntendre aux fonctions de plusieurs variables complexes la théorie des fonctions univalents. In: Montel P, ed. Lecons sur les Fonctions Univalents on Mutivalents . Paris: Gauthier-Villar, 1933
3	Chu C H. Jordan triples and Riemannian symmetric spaces. Adv Math , 2008, 219: 2029-2057 doi: 10.1016/j.aim.2008.08.001
4	Chu C H, Hamada H, Honda T, Kohr G. Distortion theorems for convex mappings on homogeneous balls. J Math Anal Appl , 2010, 369: 437-442 doi: 10.1016/j.jmaa.2010.03.014
5	Chu C H, Mellon P. Jordan structures in Banach spaces and symmetric manifolds. Expo Math , 1998, 16: 157-180
6	Gong S, Liu T S. Distortion theorems for biholomorphic convex mappings on bounded convex circular domains. Chinese Ann Math Ser B , 1999, 20: 297-304 doi: 10.1142/S0252959999000321
7	Graham I, Kohr G. Geometric Function Theory in One and Higher Dimensions. New York: Marcel Dekker, 2003
8	Graham I, Varolin D. Bloch constants in one and several variables. Pacific J Math , 1996, 174: 347-357
9	Hamada H, Honda T, Kohr G. Bohr’s theorem for holomorphic mappings with values in homogeneous balls. Israel J Math , 2009, 173: 177-187 doi: 10.1007/s11856-009-0087-9
10	Hamada H, Honda T, Kohr G. Linear invariance of locally biholomorphic mappings in the unit ball of a JB^?-triple. J Math Anal Appl , 2012, 385: 326-339 doi: 10.1016/j.jmaa.2011.06.055
11	Hamada H, Honda T, Kohr G. Trace-order and a distortion theorem for linearly invariant families on the unit ball of a finite dimensional JB^? -triple. J Math Anal Appl , 2012, 396: 829-843 doi: 10.1016/j.jmaa.2012.07.027
12	Hamada H, Honda T, Kohr G. Growth and distortion theorems for linearly invariant families on homogeneous unit balls in ?n.J Math Anal Appl , 2013, 407: 398-412 doi: 10.1016/j.jmaa.2013.05.040
13	Hamada H, Kohr G. Growth and distortion results for convex mappings in infinite dimensional spaces. Complex Var Theory Appl , 2002, 47: 291-301 doi: 10.1080/02781070290013866
14	Hamada H, Kohr G. Φ-like and convex mappings in infinite dimensional space. Rev Roumaine Math Pures Appl , 2002, 47: 315-328
15	Kaup W. A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces. Math Z , 1983, 183: 503-529 doi: 10.1007/BF01173928
16	Kaup W. Hermitian Jordan triple systems and the automorphisms of bounded symmetric domain. In: González S, ed. Non-Associative Algebra and Its Applications. Oviedo, 1993. Math Appl , Vol 303. Dordrecht: Kluwer Acad Publ, 1994 doi: 10.1007/978-94-011-0990-1_34
17	Lin Y Y, Hong Y. Some properties of holomorphic maps in Banach spaces. Acta Math Sinica (Chin Ser) , 1995, 38: 234-241 (in Chinese)
18	Liu X S, Liu T S. On the precise growth, covering, and distortion theorems for normalized biholomorphic mappings. J Math Anal Appl , 2004, 295: 404-417 doi: 10.1016/j.jmaa.2004.02.052
19	Liu X S, Liu T S. The sharp estimates for each item in the homogeneous polynomial expansions of a subclass of close-to-convex mappings. Sci Sin Math , 2010, 40: 1079-1090 (in Chinese)
20	Liu X S, Liu T S. The sharp distortion theorem for a subclass of close-to-convex mappings. Chinese Ann Math Ser A , 2012, 33: 91-100
21	Pommerenke C. On close-to-convex functions. Trans Amer Math Soc , 1965, 114: 176-186 doi: 10.1090/S0002-9947-1965-0174720-4
22	Pommerenke C. Univalent Functions. G?ttingen: Vandenhoeck & Ruprecht, 1975
23	Suffridge T J. Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions. In: Buckholtz J D, Suffridge T J, eds. Complex Analysis. Lecture Notes in Math , Vol 599. Berlin-Heidelberg-New York: Springer-Verlag, 1976, 146-159
24	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 doi: 10.1016/j.jmaa.2011.12.023
25	Zhu Y, Liu M. Distortion theorems for biholomorphic convex mappings in Banach spaces. Complex Var Theory Appl , 2005, 50: 57-68

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