The Growth of the Nevanlinna Proximity Function

Let f be a meromorphic mapping from Cn into a compact complex manifold M. In this paper we give some estimates of the growth of the proximity function mf (r,D) of f with respect to a divisor D. J.E. Littlewood [2] (cf. Hayman [1]) proved that every non-constant meromorphic function g on the complex plane C satisfies lim supr→∞ mg(r,a) log T(r,g) ≤ 1 2 for almost all point a of the Riemann sphere. We extend this result to the case of a meromorphic mapping f : Cn → M and a linear system P(E) on M. The main result is an estimate of the following type: For almost all divisor D ∈ P(E), lim supr→∞ mf (r,D)−mf (r,IB(E)) log TfE(r,HE) ≤ 1 2 .