---
tag: 30TG40QA1J
kind: definition
name: Nevanlinna characteristic
msc: ["30D35"]
defines: YT6Z0GWS0W
uses: [0NSD1TPHR0, 0WPQYW8MTE, SHF8Z7GMFX]
sources:
  - url: https://en.wikipedia.org/wiki/Nevanlinna_theory
---

## Definition

Let $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\C$, not identically zero, and write $\log^+ x = \max(\log x, 0)$ for $x \ge 0$, with $\log^+ 0 = 0$.
For $t \ge 0$, let $n(t)$ be the number of [[poles|0WPQYW8MTE]] of $f$ in the disc $\abs{z} \le t$, counted with multiplicity.
For $r > 0$, let
$$m(r, f) = \frac{1}{2\pi} \int_0^{2\pi} \log^+ \abs{f(r e^{i\theta})} \, d\theta, \qquad N(r, f) = \int_0^r \frac{n(t) - n(0)}{t} \, dt + n(0) \log r.$$
The *Nevanlinna characteristic* of $f$ is the function
$$T(r, f) = m(r, f) + N(r, f), \qquad r > 0.$$

## Remarks

Example: for $f(z) = e^z$, $\log^+ \abs{f(re^{i\theta})} = \max(r\cos\theta, 0)$ and there are no poles, so $T(r, f) = \frac{1}{2\pi}\int_{-\pi/2}^{\pi/2} r \cos\theta \, d\theta = r/\pi$.
Non-example: the maximum modulus $\max_{\abs{z} = r} \abs{f(z)}$ is no growth measure for $f(z) = \tan z$.
It is infinite whenever the circle $\abs{z} = r$ passes through a pole $\pi/2 + k\pi$.
The characteristic avoids this by counting the poles in $N(r, f)$.
The function $T(r, f)$ is increasing and a convex function of $\log r$.
For an entire function $f$ and $M(r) = \max_{\abs{z} = r} \abs{f(z)}$, one has $T(r, f) \le \log^+ M(r) \le 3\, T(2r, f)$, so $T$ and $\log M$ define the same [[order|SHF8Z7GMFX]].
A meromorphic function is rational exactly when $T(r, f) = O(\log r)$, and a rational function of degree $d$ has $T(r, f) = d \log r + O(1)$.
The Ahlfors–Shimizu characteristic, defined from the spherical area of the image of the disc $\abs{z} < r$, differs from $T(r, f)$ by a bounded function.