Functions of a complex variable Entire and meromorphic functions Theorem

Second main theorem of Nevanlinna theory

Builds on Meromorphic function Nevanlinna characteristic Nevanlinna theory Riemann sphere

Statement

Let $f$ be a meromorphic function on $\C$ that is not constant, with Nevanlinna characteristic $T(r, f)$ . Let $q \ge 3$ , and let $a_1, \dots, a_q$ be distinct points of the Riemann sphere $\C \cup \set{\infty}$ . For $a$ in the Riemann sphere, let $\overline{N}(r, a)$ be the counting function of the solutions of $f(z) = a$ counted without multiplicity, as in Nevanlinna theory. Then there is a set $E \subseteq (0, \infty)$ of finite Lebesgue measure such that $(q - 2)\, T(r, f) \le \sum_{j=1}^{q} \overline{N}(r, a_j) + S(r, f) \qquad (r \notin E),$ where $S(r, f) = O\big(\log T(r, f) + \log r\big)$ as $r \to \infty$ outside $E$ .

Remarks

The proof rests on the lemma on the logarithmic derivative, which says that the mean of $\log^+ \abs{f'/f}$ on the circle $\abs{z} = r$ is small compared with $T(r, f)$ outside an exceptional set. If $f$ is not rational, then $\log r = o(T(r, f))$ , so $S(r, f) = o(T(r, f))$ outside $E$ . If $f$ has finite order, the exceptional set $E$ can be omitted. For transcendental $f$ , dividing by $T(r, f)$ and using the first main theorem gives the defect relation $\sum_a \delta(a) \le 2$ , where the sum runs over all points of the Riemann sphere. For example, $e^z$ has $\delta(0) = \delta(\infty) = 1$ , so the relation is sharp.

Going deeper

For a student

Picard's little theorem follows at once: if a transcendental meromorphic $f$ omitted three values, the right side would be $S(r, f) = o(T(r, f))$ while the left side is at least $T(r, f)$ . The theorem also implies that the set of values $a$ with $\delta(a) > 0$ is countable. Counting without multiplicity yields results on ramification, such as bounds on the number of totally ramified values. Ahlfors theory gives a geometric proof of a similar inequality, based on covering surfaces.

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Complex numbers object
Complex plane concept
Nevanlinna characteristic concept
Nevanlinna theory concept
Real numbers object
Set bounded above definition
Real and imaginary parts definition
Riemann sphere definition
Supremum definition
Completeness axiom statement
Existence of square roots of nonnegative reals statement
Square root definition
Modulus of a complex number definition
Complex differentiability definition
Holomorphic function definition
Isolated singularity definition
Pole definition
Meromorphic function definition
Second main theorem of Nevanlinna theory statement

Tag	`GWYE6745JK`
URI	`m2://GWYE6745JK`
Kind	statement, theorem
Writing	W0 stub
MSC	30D35
Status	This copy of the site has no build data. Run `m2 status GWYE6745JK`.
About	Nevanlinna theory, Nevanlinna characteristic
Uses	Meromorphic function, Nevanlinna characteristic, Nevanlinna theory, Riemann sphere
Changed	2 November 2026