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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 $\textbf {C}^ 2$

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 $|\det {J_f}|$ and $|\arg \det {J_f}|$ 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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Article copyright: © Copyright 1994 American Mathematical Society