Growth theorem of convex mappings on bounded convex circular domains (Q1392632)

Let \(\rho\) be a \(\mathbb C\)-norm in \(\mathbb C^{n}\) and let \(\Omega:=\{z\in\mathbb C^{n}\: \rho(z)<1\}\) be the unit ball with respect to \(\rho\). Let \(f\:\Omega\rightarrow\mathbb C^{n}\) be an injective holomorphic mapping such that \(f(0)=0\), \(f'(0)=\text{id}\), and \(f(\Omega)\) is convex. The authors prove that \(\rho(z)/(1+\rho(z))\leq\rho(f(z))\leq\rho(z)/(1-\rho(z))\) for any \(z\in\Omega\). In particular, \((1/2)\Omega\subset f(\Omega)\).