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Transactions of the American Mathematical Society
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] Roger W. Barnard, Carl H. FitzGerald, and Sheng Gong, The growth and 1/4-theorems for starlike mappings in 𝐶ⁿ, Pacific J. Math. 150 (1991), no. 1, 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] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494 (85j:30034)
  • [4] Peter Duren and Walter Rudin, Distortion in several variables, Complex Variables Theory Appl. 5 (1986), no. 2-4, 323–326. MR 846500 (88c:32003)
  • [5] Sheng Gong and Zhi Min Yan, A remark on Möbius transformations. III, Chinese Quart. J. Math. 1 (1986), no. 1, 33–40 (Chinese, with English summary). MR 865350 (88e:32007c)
  • [6] Keizo Kikuchi, Starlike and convex mappings in several complex variables, Pacific J. Math. 44 (1973), 569–580. MR 0322210 (48 #572)
  • [7] Walter Rudin, Function theory in the unit ball of 𝐶ⁿ, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York-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] T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Springer, Berlin, 1977, pp. 146–159. Lecture Notes in Math., Vol. 599. MR 0450601 (56 #8894)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1250815-7
PII: S 0002-9947(1994)1250815-7
Article copyright: © Copyright 1994 American Mathematical Society