The Growth of the Nevanlinna Proximity Function

J. Math. Sci. Univ. Tokyo
Vol. 16 (2009), No. 4, Page 525--543.

Nitanda, Atsushi
The Growth of the Nevanlinna Proximity Function
Let $f$ be a meromorphic mapping from ${\bf C}^n$ into a compact complex manifold $M$. In this paper we give some estimates of the growth of the proximity function $m_f(r,D)$ 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 ${\bf C}$ satisfies $\limsup_{r \to \infty}\frac{m_g(r,a)}{\log T(r,g)} \leq \frac{1}{2}$ for almost all point $a$ of the Riemann sphere. We extend this result to the case of a meromorphic mapping $f: {\bf C}^n \to M$ and a linear system $P(E)$ on $M$. The main result is an estimate of the following type: For almost all divisor $D \in P(E)$, $\textstyle \limsup_{r \to \infty} \frac{m _{f}(r,D)-m_f (r,{\cal I}_{B(E)})}{\log T _{f_E}(r,H_E)} \leq \frac{1}{2}$.