Growth and Size of the Exceptional Set in Nevanlinna's Second Fundamental Theorem

Abstract

Let f be a transcendental meromorphic function, T(r,f) its characteristic function and S(r, f) the error term in Nevanlinna's second fundamental theorem. It is shown that for every increasing function ψ(r) such that log ψ(r) = o(r) we have S(r, f) = o(T(r, f)) outside an exceptional set E satisfying ∫E ψ(T(r, f))dr < ∞. This result makes clear the relationship between the size of the exceptional set and the growth of the characteristic function and implies that for functions of rapid lower growth improved conditions on the size of the exceptional set can be given. A general example of an entire function with a suitable exceptional set is constructed, showing that these results are essentially best possible.