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 greater than or equal to 0, the derivative f((n)) has only real zeros. We have proved that then there are real numbers a and b where a not equal 0, such that f is of the form f(az + b) = P(z)/Q(z) where Q(z) = z(n) or Q(z) = (z(2) + 1)(n) for some positive integer n, and;P is a polynomial with only real zeros such that deg P less than or equal to deg Q + 1; or f(az + b) = C(z - i)(-n) or f(az + b) = C(z - alpha)/(z - i) where alpha 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.