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Nevanlinna’s First Main Theorem with exact centre constant

Statement

Write $M_{r} h = (2 π)^{- 1} \int_{0}^{2 π} h (r e^{i t}) d t$ for a circular mean whenever it exists.

Let $f$ be a nonconstant meromorphic function on $C$ and let $a \in \hat{C}$ . For every $r > 0$ , $m (r, a; f) + N (r, a; f) = T (r, f) + C (f, a) .$ For $a = \infty$ , set $C (f, \infty) = 0$ . For finite $a$ , write the first nonzero Laurent term at the centre as $f (z) - a = c_{a} z^{k_{a}} + higher Laurent terms, c_{a} \neq 0,$ and set $C (f, a) = \frac{1}{2} \log (1 + ∣ a ∣^{2}) - \log ∣ c_{a} ∣$ . In particular, the difference $m (r, a; f) + N (r, a; f) - T (r, f)$ is $O_{f, a} (1)$ as $r \to \infty$ .

Facts & Assumptions

Given: A nonconstant meromorphic $f$ on $C$ and a target $a \in \hat{C}$ .

[F1]

The normalized chordal distance is $δ (w, a) = ∣ w - a ∣ / (\sqrt{1 + ∣ w ∣^{2}} \sqrt{1 + ∣ a ∣^{2}})$ for finite $w, a$ , and $δ (w, \infty) = 1 / \sqrt{1 + ∣ w ∣^{2}}$ (Counting, chordal proximity and characteristic).

[F2]

$m (r, a; f)$ is the circular mean of $\log (1 / δ (f, a))$ , and $T (r, f) = m (r, \infty; f) + N (r, \infty; f)$ (Counting, chordal proximity and characteristic).

[F3]

For finite $a$ , the centre-divisor Jensen identity is $M_{r} \log ∣ f - a ∣ = \log ∣ c_{a} ∣ + N (r, a; f) - N (r, \infty; f)$ , with divisor-circle means interpreted by continuous radial limits (Meromorphic Jensen identity with a zero or pole at the centre).

[F4]

The divisor counts are finite on bounded discs and $N (r, a; f)$ and $m (r, a; f)$ are finite and continuous for every $r > 0$ , including divisor radii (Well-definedness and radius conventions for Nevanlinna quantities).

Proof

technique · use the pointwise chordal identity for finite targets, average it on a regular circle, and substitute the centre-divisor Jensen identity; handle infinity directly from the definition

1.1F1given

Fix finite $a$ and a regular radius $r$ whose circle contains no pole and no $a$ -point. From [F1], at every point of that circle, $\log (1 / δ (f, a)) = \frac{1}{2} \log (1 + ∣ f ∣^{2}) + \frac{1}{2} \log (1 + ∣ a ∣^{2}) - \log ∣ f - a ∣$ .

2.1F2step 1.1

Averaging the identity in step 1.1 and using [F2] gives $m (r, a; f) - m (r, \infty; f) = \frac{1}{2} \log (1 + ∣ a ∣^{2}) - M_{r} \log ∣ f - a ∣$ .

3.1F2F3step 2.1algebra

Substitute [F3] into step 2.1 to obtain $m (r, a; f) - m (r, \infty; f) = \frac{1}{2} \log (1 + ∣ a ∣^{2}) - \log ∣ c_{a} ∣ - N (r, a; f) + N (r, \infty; f)$ . Rearranging and using [F2] yields the claimed formula for finite $a$ at each regular radius.

4.1F2F4step 3.1

Regular radii are dense because [F4] makes the divisor sets finite on bounded discs. The finite-target quantities in the formula are continuous by [F4], so the equality from step 3.1 on that dense set extends to every $r > 0$ . For $a = \infty$ , $C (f, \infty) = 0$ and the asserted identity is exactly the defining equality in [F2].

5.1step 4.1algebra∎

For each fixed $f, a$ , the constant $C (f, a)$ in step 4.1 is finite and independent of $r$ ; therefore the difference in the statement is bounded as $r \to \infty$ .

Depends on

Used by

Order and lower order from the Nevanlinna characteristic Definition
A centre a-point requires regularised counting Example
Characteristic under a target Möbius change Example
Elementary characteristic laws and fixed rational composition Theorem

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alexandre Eremenko, Lectures on Nevanlinna Theory, §1, equation (2), and §3, equations (9)–(14) (standard reference, not scraped)
Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 1 §4, Theorem 4.1 and proof (standard reference, not scraped)