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Nevanlinna deficiency and ramification index

Definition

Let $f$ be a nonconstant meromorphic function on $C$ . For a sphere target $a \in \hat{C}$ , the deficiency of $a$ is $δ (a, f) : = {lim inf}_{r \to \infty} \frac{m (r, a; f)}{T (r, f)} = 1 - {lim sup}_{r \to \infty} \frac{N (r, a; f)}{T (r, f)},$ and the ramification index of $a$ is $ε (a, f) : = {lim inf}_{r \to \infty} \frac{N_{1} (r, a; f)}{T (r, f)},$ with $N_{1} (r, a; f)$ the integrated ramification count at target $a$ , weighting each $a$ -point by its local degree minus one, as in Truncated value and ramification counts. Both indices are well-defined numbers in $[0, 1]$ (proved below).

For a set $S \subseteq \hat{C}$ of targets, the total deficiency sum is defined without choosing an enumeration by $\sum_{a \in S} (δ (a, f) + ε (a, f)) : = \sup {\sum_{a \in A} (δ (a, f) + ε (a, f)) : A \subseteq S, A finite} .$ For finite $S$ this is the ordinary finite sum. No countability of $S$ and no enumeration of $S$ is assumed.

Facts & Assumptions

Given: A nonconstant meromorphic $f$ on $C$ and a sphere target $a$ .

[F1]

$m (r, a; f)$ , $N (r, a; f)$ and $T (r, f)$ are finite for every $r > 0$ , $T$ is nondecreasing, and $T (r, f) > 1$ for all sufficiently large $r$ (Counting, chordal proximity and characteristic).

[F2]

$m (r, a; f) + N (r, a; f) = T (r, f) + C (f, a)$ with a constant $C (f, a)$ independent of $r$ ; in particular $m \geq 0$ and $N \leq T + C (f, a)$ (Nevanlinna’s First Main Theorem with exact centre constant). Integrated counts are nonnegative for $r \geq 1$ ; their centre term can be negative when $r < 1$ .

[F3]

$N (r, a; f) = \overset{ˉ}{N} (r, a; f) + N_{1} (r, a; f)$ , and $0 \leq N_{1} (r, a; f) \leq N (r, a; f)$ for $r \geq 1$ , with $N_{1} (r, a; f)$ the integrated count of $a$ -points weighted by local degree minus one (Truncated value and ramification counts).

[F4]

If $f$ is transcendental then $T (r, f) / \log r \to \infty$ , and if $f$ is rational of degree $d \geq 1$ then $T (r, f) = d \log r + O (1)$ ; in both cases $T (r, f) \to \infty$ (Transcendental characteristic dominates logarithmic growth).

Proof

technique · divide the First Main Theorem by $T$ and use that $T\to\infty$, then bound nonnegative numerators by $T$

1.1F4

$T (r, f) \to \infty$ : this is [F4] in the transcendental case, and in the rational case of degree $d \geq 1$ the formula $T (r, f) = d \log r + O (1)$ diverges.

2.1F2step 1.1algebra

Since $C (f, a)$ is a constant and $T (r, f) \to \infty$ , [F2] gives $\frac{m}{T} = 1 + \frac{C (f, a)}{T} - \frac{N}{T} = 1 - \frac{N}{T} + o (1)$ , hence ${lim inf}_{r} \frac{m}{T} = {lim inf}_{r} (1 - \frac{N}{T}) = 1 - {lim sup}_{r} \frac{N}{T}$ ; the two expressions for $δ (a, f)$ agree.

2.2F1F2step 1.1algebra

$δ (a, f) \in [0, 1]$ : from [F2], $0 \leq m \leq T + C (f, a)$ and $0 \leq N \leq T + C (f, a)$ for $r \geq 1$ ; with $T (r, f) > 0$ for all large $r$ by [F1], dividing by $T$ and taking limits gives $0 \leq lim inf \frac{m}{T}$ and $lim sup \frac{N}{T} \leq 1$ .

3.1F3step 2.2algebra

$ε (a, f) \in [0, 1]$ : for $r \geq 1$ , [F3] gives $0 \leq N_{1} (r, a; f) \leq N (r, a; f) \leq T (r, f) + C (f, a)$ ; dividing by $T$ and taking the lower limit gives $0 \leq ε (a, f) \leq lim sup \frac{N}{T} \leq 1$ .

3.2F2step 2.1algebra

If $f$ omits $a$ , then $n (t, a; f) = 0$ for every $t$ , so $N (r, a; f) \equiv 0$ and $δ (a, f) = 1 - lim sup 0 = 1$ ; also $m (r, a; f) = T (r, f) + C (f, a)$ by [F2].

4.1given∎

The sum convention is well posed as an extended nonnegative supremum: the collection is nonempty because it contains the empty subsum $0$ , and if $S$ is finite the supremum is attained at $A = S$ . No enumeration or selection is used.

Depends on

Used by

Deficiencies of the exponential and sine Example
Nevanlinna deficiency and ramification defect relations Theorem

Dependency tree · two levels

15 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

Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions (standard reference, not scraped)
I. Laine, Complex Analysis III lecture notes (standard reference, not scraped)