A SHARP FORM OF NEVANLINNAS 2ND FUNDAMENTAL THEOREM

Let f be meromorphic in the plane. We find a sharp upper bound for the error term [GRAPHICS] in Nevanlinna's second fundamental theorem. For any positive increasing functions phi(t)/t and p(t) with [GRAPHICS] and [GRAPHICS] we have S(r,f)) less-than-or-equal-to log+{phi(T(r,f))/p(r)}+O(1) as r --> infinity outside a set E with [GRAPHICS]. Further if psi(t)/t is positive and increasing and [GRAPHICS] then there is an entire f such that S(r,f) greater-than-or-equal-to log-psi(T(r,f)) outside a set of finite linear measure. We also prove analogous results for functions meromorphic in a disk.