Reality of zeros of derivatives of meromorphic functions

Let f be a meromorphic non-entire function in the plane, and suppose that for every n ≥ 0, the derivative f⁽ⁿ⁾ has only real zeros. We have proved that then there are real numbers a and b where a ≠ 0, such that f is of the form f(az+b) = P(z)/Q(z) where Q(z) = zⁿ or Q(z) = (z² + 1)ⁿ for some positive integer n , and P is a polynomial with only real zeros such that deg P ≤ deg Q + 1 ; or f(az + b) = C(z - i)^-n or f(az + b) = C(z - a)/(z - i) where α is real and C is a non-zero complex constant. In this paper we explain the structure of the proof (which is divided into several cases), and give the proof in those cases that can be dealt with by reasonably elementary methods.