Nevanlinna Theory - First Fundamental Theorem

First Fundamental Theorem

Let a ∈ C, and define

\quad N(r,a,f) = N\left(r,\dfrac{1}{f-a}\right), \quad m(r,a,f) = m\left(r,\dfrac{1}{f-a}\right).\,

For a = ∞, we set N(r,∞,f) = N(r,f), m(r,∞,f) = m(r,f).

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. For non-constant meromorphic functions in the plane, T(r, f) tends to infinity as r tends to infinity, so the First Fundamental Theorem says that the sum N(r,a,f) + m(r,a,f), tends to infinity at the rate which is independent of a. The first Fundamental theorem is a simple consequence of Jensen's formula.

The characteristic function has the following properties of the degree:

\begin{array}{lcl} T(r,fg)&\leq&T(r,f)+T(r,g)+O(1),\\ T(r,f+g)&\leq& T(r,f)+T(r,g)+O(1),\\ T(r,1/f)&=&T(r,f)+O(1),\\ T(r,f^m)&=&mT(r,f)+O(1), \, \end{array}

where m is a natural number. The bounded term O(1) is negligible when T(r,f) tends to infinity. These algebraic properties are easily obtained from Nevanlinna's definition and Jensen's formula.

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