2016/03/22 by Daniela Kraus, Oliver Roth, Kraus, Daniela +1
Mathematics · #30F45 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30F45
paper · pdf · doi:10.48550/arxiv.1603.06818
arxiv created 2016/03/22 · arxiv updated 2016/03/23
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 ℂ, which might be called Poincaré capacity. If the compact set K ⊆ 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.