A sharp form of Nevanlinna’s second fundamental theorem. (English) Zbl 0766.30025

Let \(f\) be meromorphic in the plane. R. Nevanlinna’s second fundamental theorem is a consequence of the classical estimate that \[ (*) S(r, f) \equiv m(r ,f) + \sum m(r, a_ i,f ) + N_ 1(r ,f) - 2T(r, f) \] is \(O(log rT(r, f)\) as \(r \to \infty\), avoiding a (small) \(r\)-set \(E\). In this very thorough paper, the author obtains the most precise upper bounds for \(S\), and shows how such an estimate influences the size of the associated \(E\). Similar methods give upper bounds for the logarithmic derivative \(m(r ,f'/f)\) and ramification divisor \(N_ 1(r, f) - 2T(r, f)\), as well as analogues when \(f\) is meromorphic in a finite disk.
The estimates here sharpen those of S. Lang [Lect. Notes Math. 1433, 174 p. (1990; Zbl 0709.30030)]; not only is the result simpler to state, but one of Lang’s terms can be omitted. All these results are in terms of an increasing function \(\phi (t)\) with \(\int_ 1^ \infty \{1/\phi (t)\} dt < \infty,\) and \(p(r)\) such that \(\int_ 1^ \infty \{1/p(r)\}dr = \infty.\) In the setting (*), the author proves that \[ S(r, f) \leq \log^ +\{\phi (T(r, f))/p(r)\} + const \] outside a set \(E\) for which \(\int_ E\{1/p(r)\}dr\leq 3\int_ 1^ \infty \{1/\phi (t)\} dt + const\). Thus by various choices of \(p(r)\) we obtain estimates for the linear and logarithmic measures of \(E\).
The bulk of the paper consists of examples in the plane and disk which show how close to optimal these estimates are. The exactness of the laboriously-produced constants is not considered, however.
The proof depends on a clever re-examination of R. Nevanlinna’s original argument which gave (*). As such, it is purely one-dimensional, and the analogues for several variables remain open.


30D30 Meromorphic functions of one complex variable (general theory)
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
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