The Carathéodory distance in strongly pseudoconvex domains (Q1318126)

Let \(G \Subset \mathbb{C}^ n\) be a strongly pseudoconvex domain and \(P_ 0\), \(Q_ 0 \in \partial G\). It is proved that there is a continuous double peak function \(f\) in \(G\) at \(P_ 0\), \(Q_ 0\), i.e., there exist a domain \(G' \Supset G\), two neighbourhoods \(U_ 1,U_ 2\) of \(P_ 0\) and \(Q_ 0\) respectively such that \(f:B_{U_ 1} \times B_{U_ 2} \times G' \to \mathbb{C}\): i) is continuous, ii) for every \(P \in B_{U_ 1}=U_ 1 \cap \partial G\), \(Q \in B_{U_ 2}=U_ 2 \cap \partial G\), \(f(P,Q,z)\) is a holomorphic function of \(z\) in \(G'\), iii) \(f(P,Q,P)=1\), \(f(P,Q,Q)=-1\) and \(| f(P,Q,z) |<1\), for all \(z \in \overline G-\{P,Q\}\). As a result, an estimate of the Carathéodory distance \(C_ G(z,w)\) in \(G\) is obtained for \(z,w\) sufficiently near to \(P_ 0\), \(Q_ 0\) respectively, \(P_ 0 \neq Q_ 0\).