, Volume 43, Issue 10, pp 1060-1066

The error term in Nevanlinna’s inequality

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Abstract

An upper bound is given for the error termS(r, |a j |,f) in Nevanlinna’s inequality. For given positive increasing functions p and $ with ∫ 1 dr/p(r) = ∫ 1 dr/r ϕ(r) = ∞, setP(r) = ∫ 1 r dt/p,Ψ(r) = ∫ 1 r dt/t ϕ(t) We prove that $$S(r, \left\{ {a_j } \right\}, f) \leqslant \log \frac{{T(r, f)\phi (T(r, f))}}{{p(r)}} + O(1)$$ holds, with a small exceptional set of r, for any finite set of points |a j | in the extended plane and any meromorphic function f such thatΨ(T(r, f)) =O(P(r)). This improves the known results of A. Hinkkanen and Y. F. Wang. The sharpness of the estimate is also considered.