Other main contributors in the first half of the 20th century were Lars Ahlfors, André Bloch, Henri Cartan, Edward Collingwood, Otto Frostman, Frithiof Nevanlinna, Henrik Selberg, Tatsujiro Shimizu, Oswald Teichmüller, and Georges Valiron.
The First Fundamental Theorem of Nevanlinna theory states that for every a in the Riemann sphere, where the bounded term O(1) may depend on f and a.
Then N1(r,f) is defined as the Nevanlinna counting function of critical points of f, that is The Second Fundamental theorem says that for every k distinct values aj on the Riemann sphere, we have This implies where S(r,f) is a "small error term".
Nevanlinna's original proof of the Second Fundamental Theorem was based on the so-called Lemma on the logarithmic derivative, which says that m(r,f'/f) = S(r,f).
The Second Fundamental Theorem can also be derived from the metric-topological theory of Ahlfors, which can be considered as an extension of the Riemann–Hurwitz formula to the coverings of infinite degree.
The proofs of Nevanlinna and Ahlfors indicate that the constant 2 in the Second Fundamental Theorem is related to the Euler characteristic of the Riemann sphere.
However, there is a very different explanations of this 2, based on a deep analogy with number theory discovered by Charles Osgood and Paul Vojta.
As another corollary from the Second Fundamental Theorem, one can obtain that which generalizes the fact that a rational function of degree d has 2d − 2 < 2d critical points.
Nevanlinna theory is useful in all questions where transcendental meromorphic functions arise, like analytic theory of differential and functional equations[6][7] holomorphic dynamics, minimal surfaces, and complex hyperbolic geometry, which deals with generalizations of Picard's theorem to higher dimensions.
[8] A substantial part of the research in functions of one complex variable in the 20th century was focused on Nevanlinna theory.
For example, the Inverse Problem of Nevanlinna theory consists in constructing meromorphic functions with pre-assigned deficiencies at given points.
It turns out that for this class, deficiencies are subject to several restrictions, in addition to the defect relation (Norair Arakelyan, David Drasin, Albert Edrei, Alexandre Eremenko, Wolfgang Fuchs, Anatolii Goldberg, Walter Hayman, Joseph Miles, Daniel Shea, Oswald Teichmüller, Alan Weitsman and others).
Henri Cartan, Joachim and Hermann Weyl[1] and Lars Ahlfors extended Nevanlinna theory to holomorphic curves.