A DISTORTION THEOREM FOR BIHOLOMORPHIC-MAPPINGS IN C2

Let J(f) be the Jacobian of a normalized biholomorphic mapping f from the unit ball B2 into C2. 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 B2. 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 B2 into C2 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 B2 into C2.