## 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.

### MSC:

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

### Keywords:

error term; second fundamental theorem

### Citations:

Zbl 0766.30026; Zbl 0709.30030
Full Text:

### References:

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