Bounds on the ratio $n(r, a)/S(r)$ for meromorphic functions

Abstract:

Let f be a meromorphic function in the plane. We prove the existence of an absolute constant K such that if ${a_1},{a_2}, \ldots ,{a_q}$ are distinct elements of the Riemann sphere then $\lim {\inf _{r \to \infty }}\;(\Sigma _{j = 1}^q|n(r,{a_j})/S(r) - 1|) < K$. We show by example that in general no such bound exists for the corresponding upper limit. These results involving the unintegrated functionals of Nevanlinna theory are related to previous work of Ahlfors, Hayman and Stewart, and the author.