Zeros of derivatives of meromorphic functions with one pole *

Abstract

Suppose that f (z) = g(z)/z n where g is a real entire function of finite order with g(0)≠ 0 and n is a positive integer, and that f f′, and f″ have only red zeros. We prove thatthen g is a polynomial of degree not exceeding n+1. This strengthens an earlier result of the author, when one had to consider derivatives up to an order depending on f and the order of the entire function gConversely, if f is of this form where g is a polynomial of degree at most n with only real zeros, then f(k) has only real zeros for all k≥ 0. If the degree of g is n+1 then f(k) has only real zeros for all k≥ 0 if, and only iff and f′ have only real zeros. The proof makes extensive use of complex dynamics