## 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:

 [1] Borel, E.: Sur les zéros des fonctions entières. Acta Math.20, 357–396 (1896–1897) · JFM 28.0360.01 [2] Gol’dberg, A.A., Grinshtein, V.A.: The logarithmic derivative of a meromorphic function (Russian): Mat. Zametki19, 525–530 (1976); English transl. in Math. Notes19, 320–323 (1976) [3] Hayman, W.K.: Meromorphic functions. Oxford: Clarendon Press 1964 · Zbl 0115.06203 [4] Lang, S.: Transcendental numbers and diophantine approximations. Bull. Am. Math. Soc.77, 635–677 (1971) · Zbl 0218.10053 [5] Lang, S.: The error term in Nevanlinna theory, Duke Math. J.56, 193–218 (1988). · Zbl 0659.32005 [6] Lang, S., Cherry, W.: Topics in Nevanlinna theory III, Lect. Notes Math. vol. 1433. New York: Springer, 1990 · Zbl 0709.30030 [7] Miles, J.: A sharp form of the lemma on the logarithmic derivative, to appear in J. London Math. Soc. · Zbl 0706.30023 [8] Nevanlinna, R.: Le théorème de Picard-Borel et la théorie des fonctions méromorphes. Paris, 1929. Reprinted by Chelsea, New York, 1974 · JFM 55.0773.03 [9] Nevanlinna, R.: Remarques sur les fonctions monotones, Bull. Sci. Math.55, 140–144 (1931) · JFM 57.0359.03 [10] Osgood, C.F.: Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better, J. Number Theory21, 347–389 (1985) · Zbl 0575.10032 [11] Roth, K.F.: Rational approximations to algebraic numbers, Mathematika2, 1–20 (1955) · Zbl 0064.28501 [12] Vojta, P.: Diophantine approximations and value distribution theory, Lect. Notes Math. vol. 1239. New York: Springer, 1987 · Zbl 0609.14011 [13] Wong, P.: On the second main theorem in Nevanlinna theory, Amer. J. Math.111, 549–583 (1989) · Zbl 0714.32009 [14] Ye, Z.: On Nevanlinna’s error terms, Duke Math. J.64, 243–260 (1991) · Zbl 0766.30026
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