Strong submultiplicativity of the Poincare metric

Abstract

We give a direct proof of an important result of Solynin which says that the Poincaré 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 Poincaré capacity. If the compact set $K\subseteq \mathbb{C}$ is connected, then the Poincaré 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 Poincaré metrics and universal covering maps this inequality is extended to the much wider class of universal covering maps.