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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