§14.1 Nevanlinna’s Counting, Proximity, and Characteristic Functions

14.1 Nevanlinna’s Counting, Proximity, and Characteristic Functions

Let $𝑓 : 𝐷 (0, 𝑅) \to \hat{ℂ}$ be meromorphic with a Laurent expansion

𝑓 (𝑧) = \sum_{𝑗 = 𝑘}^{\infty} 𝑐_{𝑗} 𝑧^{𝑗}

where $𝑐 = 𝑐_{𝑘}$ and $𝑐 \neq 0$ , such that the annulus of convergence has an inner radius of $0$ (we effectively take the innermost expansion). Then $𝑧 \mapsto 𝑓 (𝑧) / 𝑧^{𝑘}$ is holomorphic on a neighborhood of $0$ and is nonzero at $0$ . By Jensen’s formula (Theorem 4.4.4.2 $Theorem 4.4.4.2 Theorem 4.4.4.2$ ) on $𝑓 (𝑧) / 𝑧^{𝑘}$ ,

\begin{aligned} log | 𝑟^{𝑚 - 𝑛} \frac{𝑏_{1} 𝑏_{2} \dots 𝑏_{𝑛}}{𝑎_{1} 𝑎_{2} \dots 𝑎_{𝑚}} | + 𝑘 log 𝑟 \\ \begin{matrix}  \end{matrix} = \frac{1}{2 π} \int_{0}^{2 π} log | 𝑓 (𝑟 e^{i 𝜃}) | d 𝜃 - log | 𝑐 |, (14.1 . 1) \end{aligned}

where $𝑎_{1}, \dots, 𝑎_{𝑚}$ and $𝑏_{1}, \dots, 𝑏_{𝑛}$ are the zeros and poles of $𝑓$ in $\overline{𝐷 (0, 𝑟)} \subset 𝐷 (0, 𝑅)$ excluding those at $0$ , ordered in non-decreasing moduli. Observe that

\begin{aligned} log | \frac{𝑟^{𝑚}}{𝑎_{1} 𝑎_{2} \dots 𝑎_{𝑚}} | & = log | \frac{𝑎_{2}}{𝑎_{1}} {(\frac{𝑎_{3}}{𝑎_{2}})}^{2} \dots {(\frac{𝑎_{𝑚}}{𝑎_{𝑚 - 1}})}^{𝑚 - 1} {(\frac{𝑟}{𝑎_{𝑚}})}^{𝑚} | \\ = (\sum_{𝑗 = 1}^{𝑚 - 1} 𝑗 \int_{| 𝑎_{𝑗} |}^{| 𝑎_{𝑗 + 1} |} + 𝑚 \int_{| 𝑎_{𝑚} |}^{𝑟}) \frac{d 𝑥}{𝑥} . \end{aligned}

For any $𝑗 \in ℕ$ and $𝑥$ such that $| 𝑎_{𝑗} | \leq 𝑥 < | 𝑎_{𝑗 + 1} |$ , the number of zeros (multiplicities counted) of $𝑓$ within $\overline{𝐷 (0, 𝑥)}$ , denoted $𝑛 (𝑥, 0, 𝑓)$ , is trivially equal to $𝑛 (𝑥, 0, 𝑓) = 𝑗 + 𝑘$ if $𝑘 > 0$ , since the zero at $0$ has multiplicity $𝑘$ and the other zeros in $\overline{𝐷 (0, 𝑥)}$ are $𝑎_{1}, \dots, 𝑎_{𝑗}$ ; thus,

log | \frac{𝑟^{𝑚}}{𝑎_{1} 𝑎_{2} \dots 𝑎_{𝑚}} | = {\begin{cases} \int_{0}^{𝑟} \frac{𝑛 (𝑥, 0, 𝑓) - 𝑘}{𝑥} d 𝑥 & if 𝑘 > 0 \\ \int_{0}^{𝑟} \frac{𝑛 (𝑥, 0, 𝑓)}{𝑥} d 𝑥 & otherwise \end{cases}

In either case, this quantity is given by

log | \frac{𝑟^{𝑚}}{𝑎_{1} 𝑎_{2} \dots 𝑎_{𝑚}} | = \int_{0}^{𝑟} \frac{𝑛 (𝑥, 0, 𝑓) - 𝑛 (0, 0, 𝑓)}{𝑥} d 𝑥 .

Letting $𝑛 (𝑥, \infty, 𝑓)$ denote the number of poles, counting orders, in $\overline{𝐷 (0, 𝑥)}$ , we obtain similarly

log | \frac{𝑟^{𝑛}}{𝑏_{1} 𝑏_{2} \dots 𝑏_{𝑛}} | = \int_{0}^{𝑟} \frac{𝑛 (𝑥, \infty, 𝑓) - 𝑛 (0, \infty, 𝑓)}{𝑥} d 𝑥 .

Definition 14.1.1.

For $𝑎 \in \hat{ℂ}$ , define the Nevanlinna counting function by

𝑁 (𝑟, 𝑎, 𝑓) = \int_{0}^{𝑟} \frac{𝑛 (𝑥, 𝑎, 𝑓) - 𝑛 (0, 𝑎, 𝑓)}{𝑥} d 𝑥 + 𝑛 (0, 𝑎, 𝑓) log 𝑟,

where $𝑛 (𝑥, 𝑎, 𝑓)$ counts the number of times $𝑓$ attains $𝑎$ in $\overline{𝐷 (0, 𝑥)}$ , counting orders (multiplicities of zeros of $𝑓 - 𝑎$ , or poles of $𝑓$ if $𝑎 = \infty$ ). In the event that $𝑎$ is elided, assume $𝑎 = \infty$ .

Then (14.1.1) $(14.1.1) (14.1.1)$ is simply

𝑁 (𝑟, 0, 𝑓) - 𝑁 (𝑟, \infty, 𝑓) = \frac{1}{2 π} \int_{0}^{2 π} log | 𝑓 (𝑟 e^{i 𝜃}) | d 𝜃 - log | 𝑐 | . (14.1 . 2)

Define the nonnegative part of the logarithm, denoted ${log}^{+} (𝑥)$ , to be

{log}^{+} (𝑥) = max {log (𝑥), 0} .

Proposition 14.1.2 (Properties of ${log}^{+}$ ).

The following properties hold:

${log}^{+} 𝑥 \geq log 𝑥$ for $𝑥 > 0$ .
${log}^{+} 𝑥 \leq {log}^{+} 𝑦$ for $𝑥 \leq 𝑦$ (non-decreasing).
$log 𝑥 = {log}^{+} 𝑥 - {log}^{+} (\frac{1}{𝑥})$ for $𝑥 > 0$ .
$| log 𝑥 | = {log}^{+} 𝑥 + {log}^{+} (\frac{1}{𝑥})$ for $𝑥 > 0$ .
${log}^{+} \prod_{𝑘} 𝑥_{𝑘} \leq \sum_{𝑘} {log}^{+} 𝑥_{𝑘}$ for (finitely many) positive $𝑥_{𝑘}$ .
${log}^{+} \sum_{𝑘 = 1}^{𝑛} 𝑥_{𝑘} \leq log 𝑛 + \sum_{𝑘 = 1}^{𝑛} {log}^{+} 𝑥_{𝑘}$ for (finitely many) positive $𝑥_{𝑘}$ .

Proof.

The first five properties are trivial. We now prove Part 6 $Part 6 Part 6$ . Observing that $𝑥_{𝑘} \leq e^{{log}^{+} 𝑥_{𝑘}}$ , it follows that

\begin{aligned} {log}^{+} \sum_{𝑘 = 1}^{𝑛} 𝑥_{𝑘} & \leq {log}^{+} \sum_{𝑘 = 1}^{𝑛} e^{{log}^{+} 𝑥_{𝑘}} \leq {log}^{+} 𝑛 + {log}^{+} (e^{{max}_{𝑘} {log}^{+} 𝑥_{𝑘}}) \\ = log 𝑛 + max_{𝑘} {log}^{+} 𝑥_{𝑘} . \end{aligned}

The last step uses the fact that ${log}^{+}$ is simply $log$ when the argument $\geq 1$ . ⁠ $□$

Definition 14.1.3.

For $𝑟 > 0$ , $𝑎 \in ℂ$ , define the Nevanlinna proximity function by

𝑚 (𝑟, 𝑎, 𝑓) = \frac{1}{2 π} \int_{0}^{2 π} {log}^{+} | \frac{1}{𝑓 (𝑟 e^{i 𝜃}) - 𝑎} | d 𝜃

and

𝑚 (𝑟, 𝑓) = 𝑚 (𝑟, \infty, 𝑓) = \frac{1}{2 π} \int_{0}^{2 π} {log}^{+} | 𝑓 (𝑟 e^{i 𝜃}) | d 𝜃

in the infinite case. Observe that $𝑚 (𝑟, 𝑎, 𝑓) = 𝑚 (𝑟, \frac{1}{𝑓 - 𝑎})$ .

Since

{log}^{+} | 𝑓 (𝑟 e^{i 𝜃}) | - {log}^{+} | \frac{1}{𝑓 (𝑟 e^{i 𝜃})} | = log | 𝑓 (𝑟 e^{i 𝜃}) | .

this implies

𝑚 (𝑟, \infty, 𝑓) - 𝑚 (𝑟, 0, 𝑓) = \frac{1}{2 π} \int_{0}^{2 π} log | 𝑓 (𝑟 e^{i 𝜃}) | d 𝜃 .

Then (14.1.2) $(14.1.2) (14.1.2)$ can be written as:

𝑁 (𝑟, 0, 𝑓) + 𝑚 (𝑟, 0, 𝑓) = 𝑁 (𝑟, \infty, 𝑓) + 𝑚 (𝑟, \infty, 𝑓) - log | 𝑐 | . (14.1 . 3)

Definition 14.1.4.

This motivates the sum

𝑇 (𝑟, 𝑓)

, known as the Nevanlinna characteristic, defined by

𝑇 (𝑟, 𝑓) = 𝑚 (𝑟, 𝑓) + 𝑁 (𝑟, 𝑓)

(we will also use the notation

𝑇 (𝑟, 𝑎, 𝑓) = 𝑚 (𝑟, 𝑎, 𝑓) + 𝑁 (𝑟, 𝑎, 𝑓)

Then (14.1.3) $(14.1.3) (14.1.3)$ can be rewritten as:

Proposition 14.1.5.

For any (non-constant) meromorphic $𝑓$ on $𝐷 (0, 𝑅)$ with the innermost Laurent series $𝑓 (𝑧) = \sum_{𝑗 = 𝑘}^{\infty} 𝑐_{𝑗} 𝑧^{𝑗}$ and any $0 < 𝑟 < 𝑅$ ,

𝑇 (𝑟, 0, 𝑓) = 𝑇 (𝑟, \infty, 𝑓) - log | 𝑐_{𝑘} | .