Quotient representations of meromorphic functions

Abstract

Let f be a meromorphic function in [z I 0. It is implicit in the method of proof that for any B > 1 there is a corresponding A for which the desired representation holds for all f . We show that in general B cannot be chosen to be 1 by giving an example of a meromorphicfsuch that if f = fl]f2 where f l and f2 are entire then T(r,f2) # O(T(r,f)). Rubel and Taylor have obtained the above theorem for special classes of meromorphic functions. In particular it is shown in I-5] that such a representation exists for any meromorphic f such that either sup T(2r,f)]T(r,f) _l or such that log T(r,f) is a convex function of log r. Although the representation is not obtained in [-5] for all meromorphic functions, the general result is shown to be equivalent to a seemingly more elementary proposition concerning sequences of complex numbers. The contribution of this paper is to prove the result concerning sequences of complex numbers and to provide an example showing the theorem is sharp. Results in this direction for functions of several complex variables appear