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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A distortion theorem for biholomorphic mappings in $\textbf {C}^ 2$
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by Roger W. Barnard, Carl H. FitzGerald and Sheng Gong PDF
Trans. Amer. Math. Soc. 344 (1994), 907-924 Request permission


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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 344 (1994), 907-924
  • MSC: Primary 32H02
  • DOI:
  • MathSciNet review: 1250815