Generalized logarithmic derivative estimates of Gol'dberg-Grinshtein type

For f meromorphic in the complex plane and W meromorphic in the unit disc, sharp upper bounds are obtained for m (r, f((k))/f((j))) = 1/2pi integral(0)(2pi) log(+) \f((k))(re(itheta))/f((j))(re(itheta))\ dtheta, r < infinity and m (r, phi((k))/phi((j))) = 1/2pi integral(0)(2pi) log(+) \phi((k))(re(itheta))/phi((j))(re(itheta))\ dtheta, r < 1 where k and j are integers satisfying k > j greater than or equal to 0. The results generalize the logarithmic derivative estimate due to Gol'dberg and Grinshtein to derivatives of higher order.