Abstract. Let $\kappa $ be a positive integer. A sequence $(f_n)$ of generalized Nevanlinna functions of the class ${\bf N}_\kappa $, 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 $\bf N_{\kappa '}$ with $0\le\kappa '\le\kappa $. In this note we show that if it is a priori known that $f$ belongs to the same class ${\bf N}_\kappa $ then the sequence $(f_n)$ converges locally uniformly on the set $({\msbm C}\setminus{\msbm R})\cap{\rm hol}f$, and the sets of poles or generalized poles of nonpositive type of $f_n$ converge to the set of poles or generalized poles of nonpositive type of $f$. Moreover, a compactness result for generalized Nevanlinna functions is proved.

AMS Subject Classification (1991): 30E20, 30C15, 46C20, 46G99

Keyword(s): generalized Nevanlinna functions, rational functions, locally uniform convergence

Received February 25, 2011, and in revised form July 8, 2011. (Registered under 12/2011.)