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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A distortion theorem for biholomorphic mappings in $ {\bf C}\sp 2$

Authors: Roger W. Barnard, Carl H. FitzGerald and Sheng Gong
Journal: Trans. Amer. Math. Soc. 344 (1994), 907-924
MSC: Primary 32H02
MathSciNet review: 1250815
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Abstract: Let $ {J_f}$ be the Jacobian of a normalized biholomorphic mapping f from the unit ball $ {B^2}$ into $ {\mathbb{C}^2}$. An expression for the $ \log \det {J_f}$ is determined by considering the series expansion for the renormalized mappings F obtained from f under the group of holomorphic automorphisms of $ {B^2}$. This expression is used to determine a bound for $ \vert\det {J_f}\vert$ and $ \vert\arg \det {J_f}\vert$ for f in a compact family X of normalized biholomorphic mappings from $ {B^2}$ into $ {\mathbb{C}^2}$ in terms of a bound $ C(X)$ of a certain combination of second-order coefficients. Estimates are found for $ C(X)$ for the specific family X of normalized convex mappings from $ {B^2}$ into $ {\mathbb{C}^2}$.

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  • [1] R. W. Barnard, C. H. Fitzgerald, and Sheng Gong, The growth and $ 1/4$-theorems for starlike functions in $ {\mathbb{C}^n}$, Pacific J. Math. 150 (1991), 13-22. MR 1120709 (92g:32046)
  • [2] H. Cartan, Sur la possibilité d'étendre sux fonctions de plusieurs variables complexes la théorie des fonctions univalentes, Note added to P. Montel, Leçons sur les fonctions univalentes on multivalentes, Gauthier-Villars, Paris, 1993, pp. 129-155.
  • [3] P. Duren, Univalent functions, #259, Springer-Verlag, New York, 1983. MR 708494 (85j:30034)
  • [4] P. Duren and W. Rudin, Distortion in several variables, Complex Analysis: Theory & Appl. 5 (1986), 323-326. MR 846500 (88c:32003)
  • [5] S. Gong and Z. M. Yan, A remark on Möbius transformations (III), Chinese Quart. J. 1 (1987), 33-40. MR 865350 (88e:32007c)
  • [6] K. Kikuchi, Starlike and convex mappings in several complex variables, Pacific J. Math. 44 (1973), 569-580. MR 0322210 (48:572)
  • [7] W. Rudin, Function theory in the unit ball of $ {\mathbb{C}^n}$, Grundlehren Math. Wiss., Springer, Berlin, 1980. MR 601594 (82i:32002)
  • [8] T. J. Suffridge, The principle of subordination applied to functions of several variables, Pacific J. Math. 33 (1970), 241-248. MR 0261040 (41:5660)
  • [9] -, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Lecture Notes in Math., vol. 599, Springer-Verlag, New York, 1976, pp. 146-159. MR 0450601 (56:8894)

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Article copyright: © Copyright 1994 American Mathematical Society

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