Convex and Starlike Mappings in Several Complex Variables by Sheng GongConvex and Starlike Mappings in Several Complex Variables by Sheng Gong

Convex and Starlike Mappings in Several Complex Variables

bySheng Gong

Paperback | November 6, 2012

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This interesting book deals with the theory of convex and starlike biholomorphic mappings in several complex variables. The underly­ ing theme is the extension to several complex variables of geometric aspects of the classical theory of univalent functions. Because the author's introduction provides an excellent overview of the content of the book, I will not duplicate the effort here. Rather, I will place the book into historical context. The theory of univalent functions long has been an important part of the study of holomorphic functions of one complex variable. The roots of the subject go back to the famous Riemann Mapping Theorem which asserts that a simply connected region n which is a proper subset of the complex plane C is biholomorphically equivalent to the open unit disk <. that="" _is2c_="" there="" is="" a="" univalent="" function="" _28_holoc2ad_="" morphic="" _bijection29_="" i="" _3a_=""><_20_-2b_20_n.20_in20_the20_early20_part20_of20_this20_century20_work20_began20_to20_focus20_on20_the20_class20_s20_of20_normalized20_28_f20_28_029_20_3d_20_020_and20_i27_27_20_28_029_20_3d_20_129_20_univalent20_functions20_defined20_on20_the20_unit20_disk.20_the20_restriction20_to20_unic2ad_20_valent20_functions20_defined20_on20_the20_unit20_disk20_is20_justified20_by20_the20_riemann20_mapping20_theorem.20_the20_subject20_contains20_many20_beautiful20_results20_that20_were20_obtained20_by20_fundamental20_techniques20_developed20_by20_many20_mathec2ad_20_maticians2c_20_including20_koebe2c_20_bieberbach2c_20_loewner2c_20_goluzin2c_20_grunsky2c_20_and20_schiffer.20_the20_best-known20_aspect20_of20_univalent20_function20_theory20_is20_the20_so-called20_bieberbach20_conjecture20_which20_was20_proved20_by20_de20_branges20_in20_1984. _-2b_="" n.="" in="" the="" early="" part="" of="" this="" century="" work="" began="" to="" focus="" on="" class="" s="" normalized="" _28_f="" _28_029_="1)" and="" _i27_27_="" univalent="" functions="" defined="" unit="" disk.="" restriction="" _unic2ad_="" valent="" disk="" is="" justified="" by="" riemann="" mapping="" theorem.="" subject="" contains="" many="" beautiful="" results="" that="" were="" obtained="" fundamental="" techniques="" developed="" _mathec2ad_="" _maticians2c_="" including="" _koebe2c_="" _bieberbach2c_="" _loewner2c_="" _goluzin2c_="" _grunsky2c_="" schiffer.="" best-known="" aspect="" function="" theory="" so-called="" bieberbach="" conjecture="" which="" was="" proved="" de="" branges="">
Title:Convex and Starlike Mappings in Several Complex VariablesFormat:PaperbackDimensions:209 pagesPublished:November 6, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401061912

ISBN - 13:9789401061919

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Table of Contents

Introduction: Introduction. Counterexamples. I. Criteria for Starlikeness for Holomorphic Mappings. II. Criteria for Convexity for Holomorphic Mappings. III. The Growth Theorem for Holomorphic Starlike Mappings. IV. The Growth Theorem for Holomorphic Convex Mappings. V. The Distortion Theorem for the Linear-invariant Family. VI. The Distortion Theorem for Holomorphic Convex and Starlike Mappings. VII. The Geometrical Properties for Holomorphic Convex Mappings on the Unit Ball. References. List of Symbols. Index.