A sharp form of Nevanlinna's second fundamental theorem

Let f be meromorphic in the plane. We find a sharp upper bound for the error term {Mathematical expression} in Nevanlinna's second fundamental theorem. For any positive increasing functions φ{symbol}(t)/t and p(t) with {Mathematical expression} and {Mathematical expression} we have {Mathematical expression} as r→∞ outside a set E with {Mathematical expression}. Further if ψ(t)/t is positive and increasing and {Mathematical expression} then there is an entire f such that S(r, f)≧logψ(T(r, f)) outside a set of finite linear measure. We also prove analogous results for functions meromorphic in a disk.