Regions of meromorphy and value distribution of geometrically converging rational functions

Let D be a region, { r n } n ∈ N a sequence of rational functions of degree at most n and let each r n have at most m poles in D, for m ∈ N fixed. We prove that if { r n } n ∈ N converges geometrically to a function f on some continuum S ⊂ D and if the number of zeros of r n in any compact subset of D is of growth o ( n ) as n → ∞ , then the sequence { r n } n ∈ N converges m 1 -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 m 1 -maximally convergent rational functions, especially in Padé approximation and Chebyshev rational approximation.