A distortion theorem for biholomorphic mappings in \({\mathbb{C}}^ 2\) (Q923168)

The main result of this paper is the following: Let S be a space invariant under biholomorphic automorphisms of the unit ball of \({\mathbb{C}}^ 2\). Let \(X\subset S\) consist of normalized mappings \(f=(f_ 1,f_ 2)\) where \[ f_ 1(z_ 1,z_ 2)=z_ 1+d^{(1)}_{2,0}z^ 2_ 1+d^{(1)}_{1,1}z_ 1z_ 2+d^ 1_{0,2}z^ 2_ 2+..., \] \[ f_ 2(z_ 1,z_ 2)=z_ 2+d^{(2)}_{2,0}z^ 2_ 1+d^{(2)}_{1,1}z_ 1z_ 2+d^{(2)}_{0,2}z^ 2_ 2+.... \] Let J be the Jacobian of f at z and let \(C(X)=\sup [| d^{(1)}_{2,0}+d^{(2)}_{1,1}|:\) \(f\in X].\) If C(X) is finite, then \[ | \log [1-z\bar z)^{3/2}\det J]| \leq C(X)\log (1+| z|)/(1-| z|)\text{ where } | z|^ 2=z\bar z\text{ for } z=(z_ 1,z_ 2). \] Further, if S is a space of biholomorhic convex mappings and K is a subset of normalized convex mappings, then \(C(k)<1.760..\). If \(f\in K\), then it is conjectured that \[ | \log [(1-z\bar z)^{3/2}\det J_ f]\leq (3/2)\log (1+| z|)/(1-| z|). \] It is stated that these results can be extended to \({\mathbb{C}}^ n\).