The second derivative of a meromorphic function

Let f be meromorphic of finite order in the plane, such that f(k) has finit ely many zeros, for some k greater than or equal to 2. The author has conje ctured that f then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of f. Further, we sho w that the conjecture is true if either (i) f has order less than 1 + epsilon, for some positive absolute constant epsilon, or (ii) f((m)), for some 0 less than or equal to m < k, has few zeros away fro m the real axis.