A Schwarz lemma for convex domains in arbitrary Banach spaces (Q1917665)

The classical Schwarz lemma which states \(|f(z)|\leq|z|\) for all \(z\in\mathbb{D}\) and all holomorphic functions \(f\in{\mathcal H}(\mathbb{D})\) with \(f(0)=0\) is generalized to holomorphic functions on bounded convex domains in complex Banach spaces. As the main result the following is shown: Theorem: Let \({\mathcal D}_1\subseteq E\) and \({\mathcal D}_2\subseteq F\) be two convex domains in some complex Banach spaces \(E\) and \(F\) and suppose that \({\mathcal D}_1\) is bounded. Then for each \(a\in{\mathcal D}_1\), \(b\in{\mathcal D}_2\) and \(r>0\) there is some number \(s=s(a,b,r)\) such that \(\text{dist}(z,{\mathcal D}^c_1)>r\) implies \(\text{dist} (f(z),{\mathcal D}^c_2)>s\) for all \(z\in{\mathcal D}_1\) and all holomorphic functions \(f\in{\mathcal H}({\mathcal D}_1,{\mathcal D}_2)\) satisfying \(f(a)=b\). There is an explicit formula for \(s(a,b,r)\) expressed in terms of the norms of \(E\) and \(F\). The proof is based on the Schwarz-Pick inequality for the Carathéodory pseudodistances associated to \({\mathcal D}_1\) and \({\mathcal D}_2\), respectively, combined with a relation of that distance to the norm distance.