Distortion theorems for biholomorphic convex mappings on bounded convex circular domains
The main purpose of this paper is to generalize certain distortion theorems for convex mappings by considering bounded convex circular domains instead of the unit ball. Let \(p=p(z)\) be the Minkowski functional of the domain, and let \(F=F(z,v)\) be the infinitesimal form of the Kobayashi (= Carathéodory) metric. The auhtors give the sharp upper and lower bounds for \(p(J_f(z)v)\) in terms of \(p(z)\) and \(F(z,v)\), where \(J_f\) denotes the jacobian matrix. This generalizes the corresponding estimate on the ball proved by \textit{S. Gong, S. Wang} and \textit{Q. Yu} [Pac. J. Math. 161, No. 2, 287-306 (1993; Zbl 0788.32017)]. In addition, Gong and Liu derive in terms of \(p\), upper and lower bounds for the eigenvalues of the jacobian. The question remains open as to the sharp upper and lower bounds for \(|\det J_f|\) for convex mappings, even on the ball.
scientific article; zbMATH DE number 1385314