Sharp forms of nevanlinna’s error terms

Abstract

Let f(z) be a meromorphic function in the plane. If ψ(t)/t andp(t) are two positive, continuous and non-decreasing functions on [1,∞) with ∫ ^∞₁ dt/ψ(t) = ∞ and ∫ ^∞₁ dt/p(t) = ∞, then$S(r,f) \le \log + \frac{{\psi \left( {T(r,f)} \right)}}{{p(r)}} + O(1)$ asr → ∞ outside a small exceptional set, provided that the divergence of the integral ∫ ^r₁ dt/ψ(t) is slow enough. The same forms for the logarithmic derivative and for the ramification term are obtained. It is shown by example that the estimates are best possible.