Convergence in capacity of rational approximants of meromorphic functions

Abstract

Let f be meromorphic on the compact set E ? C with maximal Green domain of meromorphy Ep(f), p(f) < ?. We investigate rational approximants with numerator degree ? n and denominator degree ? mn for f. We show that the geometric convergence rate on E implies convergence in capacity outside E if mn = o(n) as n ? 1. Further, we show that the condition is sharp and that the convergence in capacity is uniform for a subsequence ? ? N.