Strong submultiplicativity of the Poincaré metric

Abstract

We give a direct proof of an important result of Solynin which says that the Poincare metric is a strongly submultiplicative domain function. This result is then used to define a new capacity for compact subsets of the complex plane $${\mathbb {C}}$$ , which might be called Poincare capacity. If the compact set $$K \subseteq {\mathbb {C}}$$ is connected, then the Poincare capacity of K is the same as the logarithmic capacity of K. In this special case, the submultiplicativity is well-known and can be stated as an inequality for the normalized conformal map onto the complement of K. Using the connection between Poincare metrics and universal covering maps this inequality is extended to the much wider class of universal covering maps.