Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, {rn}n∈N a sequence of rational functions of degree at most n and let each rn have at most m poles in D, for m∈N fixed. We prove that if {rn}n∈N converges geometrically to a function f on some continuum S⊂D and if the number of zeros of rn in any compact subset of D is of growth o(n) as n→∞, then the sequence {rn}n∈N converges m1-almost uniformly to a meromorphic function in D. This result about meromorphic continuation is used to obtain Picard-type theorems for the value distribution of m1-maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.