First main theorem of Nevanlinna theory

Remarks

Apply Jensen's formula to $g = f - a$. In Nevanlinna's notation it reads $T(r, g) - T(r, 1/g) = \log \abs{g(0)}$, and $T(r, 1/g) = m(r, a) + N(r, a)$. The inequalities $\log^+ \abs{u - a} \le \log^+ \abs{u} + \log^+ \abs{a} + \log 2$ and its symmetric form show that $T(r, f - a)$ and $T(r, f)$ differ by at most $\log^+ \abs{a} + \log 2$. For example, for $f(z) = e^z$ and $a = 0$ there are no solutions of $f(z) = 0$, so $N(r, 0) = 0$, and $m(r, 0) = r/\pi = T(r, f)$.

Going deeper

For a student

The first main theorem is a reformulation of Jensen's formula, and it is easy compared with the second main theorem. It gives the upper bound $N(r, a) \le T(r, f) + O(1)$, which says that no value is taken too often. It also justifies the definition of the defect $\delta(a) = 1 - \limsup_{r \to \infty} N(r, a)/T(r, f)$, which lies between $0$ and $1$. The second main theorem gives the much deeper lower bound for most values.