On moduli of boundary values of holomorphic functions (Q1189276)

An interesting result is proved here: Let \(\Delta\) denote the unit disc in \(\mathbb{C}\) and let \(\varphi\) be a nonnegative continuous function on \(b\Delta\) such that \(\log\varphi\) is integrable. Let \(\|\cdot\|\) be a norm on \(\mathbb{C}^ 2\) for which the unit ball is strictly convex. Then there is a continuous map \(f:\bar\Delta\to\mathbb{C}^ 2\) which is holomorphic on \(\Delta\) such that \(\| f(\zeta)\|=\varphi(\zeta)\) for all \(\zeta\in b\Delta\). The corresponding statement for \(f\) in the disc algebra is false. More generally if \(D\) is a bounded strictly convex domain in \(\mathbb{C}^ 2\) containing 0 one can obtain \(f(\zeta)\in b(\varphi(\zeta)D)\) for all \(\zeta\in b\Delta\).