A topological generalization of Schwarz' lemma

The classical lemma of Schwarz states that if f(a) = zg(z) where g(z) is holomorphic inside the unit circle, and if |f(z)| < 1 when |s| < 1, then |f(z)| < |z| when |z| < 2. The proof for this is based on the fast that if f(s) is holomorphic in a region, its absolute value has no relative maxima in the interior of the region. Max Zorn has stated and proved a more general, highly axiomatic version of Schwarz' lemma, applicable to certaln families of transformations of a metrizable topological space into itself. But a geometrical interpretation of Zorn*s results is impossible without severe additional restrictions on the space and the families of transformations. The present paper proves a geometrical version of Schwarz' lemma by metrical-topological methods. The proof is applicable to the euclidean KL -space in particular, and, more generally, to any convex topological space provided each point of the space lies on one of the surfaces of a system of concentric compact spheres