Convergence of generalized Nevanlinna functions

Abstract

Let κ be a positive integer. A sequence (fn) of generalized Nevanlinna functions of the class Nκ, which converges locally uniformly on some nonempty open subset of the complex plane to a function f, need not converge on any larger set, and f can belong to any class Nκ’ with 0≤ κ’ ≤ κ. In this note we show that if it is a priori known that f belongs to the same class Nκ then the sequence (fn) converges locally uniformly on the set (ℂℝ)ȩholf, and the sets of poles or generalized poles of nonpositive type of fn converge to the set of poles or generalized poles of nonpositive type of f. Moreover, a compactness result for generalized Nevanlinna functions is proved.