# Nevanlinna theorems

Two fundamental theorems, proved by R. Nevanlinna (see [1], [2]), that are basic for the theory of value distribution of meromorphic functions (see Value-distribution theory). Let be a meromorphic function on a disc

where means that is meromorphic in the entire open complex plane. For every , , the proximity function of to a number is defined by

and the counting function of the number of -points of by

where denotes the number of -points of , counting multiplicities, in the disc , i.e. the number of elements of , and for , for .

The function is called the Nevanlinna characteristic of .

Nevanlinna's first theorem. For any function that is meromorphic on a disc , for any , , and any complex number ,

 (1)

where

Here denotes the first non-zero coefficient in the Laurent expansion about zero of the function if , and of itself if . Thus, for a function whose characteristic increases without limit as , the sum , considered for different values of , is equal to the value up to a bounded additive term . In this sense, all values are equivalent for any function that is meromorphic on . For this reason, the theory of value distribution of meromorphic functions concerns itself with questions about the asymptotic behaviour of one term, or , in the invariant sum (1).

Nevanlinna's second theorem shows that, for almost all points , the principal role in the sum (1) is played by . The statement of the theorem is as follows.

For any function that is meromorphic on a disc , every , , and any distinct numbers in the extended complex plane, the relation

 (2)

holds, where

and the term has the following properties:

1) If , i.e. if is meromorphic in the entire open complex plane, then

as , for all values of with the possible exception of a set of finite total measure.

2) If , then

as , for all values of with the possible exception of a set for which

The function is non-decreasing with increasing , and therefore the right-hand term in (2) cannot increase as more rapidly than outside some exceptional set .

#### References

 [1] R. Nevanilinna, "Analytic functions" , Springer (1970) (Translated from German) [2] R. Nevanlinna, "Le théorème de Picard–Borel et la théorie des fonctions méromorphes" , Gauthier-Villars (1929) [3] H. Weyl, "Meromorphic functions and analytic curves" , Princeton Univ. Press (1943) [4] L. Ahlfors, "The theory of meromorphic curves" Acta Soc. Sci. Fennica. Nova Ser. A , 3 : 4 (1941) pp. 1–31 [5] H. Cartan, "Sur les zéros des combinations linéares de fonctions holomorphes données" Mathematica (Cluj) , 7 (1933) pp. 5–31 [6] P. Griffiths, J. King, "Nevanlinna theory and holomorphic mappings between algebraic varieties" Acta Math. , 130 (1973) pp. 145–220 [7] V.P. Petrenko, "The growth of meromorphic functions" , Khar'kov (1978) (In Russian)