A sharp form of Nevanlinna's second fundamental theorem

Summary

Letf be meromorphic in the plane. We find a sharp upper bound for the error term

$$S(r,f) = m(r,f) + \sum\limits_{i = 1}^q {m(r,a_i ,f)} + N_1 (r,f) - 2T(r,f)$$

in Nevanlinna's second fundamental theorem. For any positive increasing functions ϕ(t)/t andp(t) with$\int\limits_1^\infty {dt/\varphi (t)}< \infty$ and$\int\limits_1^\infty {dt/p(t)} = \infty$ we have

$$S\left( {r,f} \right) \leqq \log ^ + \left\{ {\frac{{\varphi \left( {T\left( {r,f} \right)} \right)}}{{p\left( r \right)}}} \right\} + O\left( 1 \right)$$

asr→∞ outside a setE with$\int\limits_E {dr/p(r)}< \infty$. Further if ψ(t)/t is positive and increasing and$\int\limits_1^\infty {dt/} \psi (t) = \infty$ then there is an entiref such thatS(r, f)≧logψ(T(r, f)) outside a set of finite linear measure. We also prove analogous results for functions meromorphic in a disk.