Let f be a transcendental meromorphic function, and let k and j be integers that satisfy . In this paper we prove upper estimates for which hold for where G is some specified set of points. For examples, we state the following two corollaries of our results.
Corollary 1: Let f have finite order , and let be a given constant. Then there exists a set that has linear measure zero, such that if , then there is a constant such that for all z satisfying arg z and , we have
Corollary 2: Let f have finite order , and let be a given constant. Then there exists a set that has finite logarithmic measure, such that for all z satisfying , we have
The gamma function gives examples which show that the estimates in Corollaries 1 and 2 are both sharp. We also prove an estimate (Corollary 3) like Corollary 2 in the case when the exceptional set E has finite linear measure, and this estimate turns out to be sharp by examples of Hayman. The four general theorems in the paper apply to infinite order functions as well, since these general theorems are stated in terms of the Nevanlinna characteristic and the counting functions of zeros and poles. One of the four theorems is sharp up to a constant factor. Estimates like those in this paper have countless applications in meromorphic function theory and in algebraic differential equations.