A distortion theorem for biholomorphic mappings in $\textbf {C}^ 2$

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 ${\mathbb {C}^2}$.