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Order and lower order from the Nevanlinna characteristic

Statement

Let $f$ be nonconstant meromorphic on $C$ . Its characteristic is nondecreasing, and $T (r, f) > 1$ for all sufficiently large $r$ . Define its order and lower order by $ρ (f) = {lim sup}_{r \to \infty} \frac{\log T (r, f)}{\log r}, λ (f) = {lim inf}_{r \to \infty} \frac{\log T (r, f)}{\log r},$ using only sufficiently large $r > 1$ with $T (r, f) > 1$ . Both values lie in $[0, \infty]$ ; $f$ has finite order when $ρ (f) < \infty$ . For every constant finite map, set $ρ (f) = λ (f) = 0$ by convention. For entire functions, the next proposition compares this order with the classical maximum-modulus order; meromorphic growth is measured by $T$ , which remains finite in the presence of poles.

Facts & Assumptions

Given: The characteristic, integrated counts, and normalized chordal proximity for a meromorphic function on $C$ .

[F1]

The integrated count is $N (r, a; f) = n (0, a; f) \log r + \int_{0}^{r} (n (t, a; f) - n (0, a; f)) d t / t$ (Counting, chordal proximity and characteristic).

[F2]

The characteristic is $T (r, f) = m (r, \infty; f) + N (r, \infty; f)$ (Counting, chordal proximity and characteristic).

[F3]

The normalized chordal sphere has diameter one, so $\log (1 / δ (f, a)) \geq 0$ and every proximity is nonnegative (Counting, chordal proximity and characteristic).

[F4]

The First Main Theorem gives $m (r, a; f) + N (r, a; f) = T (r, f) + C (f, a)$ with a fixed finite centre constant (Nevanlinna’s First Main Theorem with exact centre constant).

[F5]

The Ahlfors–Shimizu identity is $T (r, f) = T_{A S} (r, f) + C_{\infty} (f)$ , and $T_{A S}$ is finite and nondecreasing (Ahlfors–Shimizu area form of the characteristic).

Proof

technique · use a central divisor to force logarithmic growth of $T$, then use the area identity to establish monotonicity of the characteristic

1.1F1algebra

For any target $a$ , $n (t, a; f) \geq n (0, a; f)$ , so [F1] gives $N (r, a; f) \geq n (0, a; f) \log r$ for $r > 1$ .

2.1F2F3F4step 1.1algebra

If $f (0)$ is finite, take $a = f (0)$ ; then $n (0, a; f) \geq 1$ and [F4], [F3], and step 1.1 give $T (r, f) \geq \log r - C (f, a)$ . If $0$ is a pole, then $n (0, \infty; f) \geq 1$ and [F2], [F3], and step 1.1 give $T (r, f) \geq \log r$ . Thus $T (r, f) \to \infty$ and is greater than $1$ for all sufficiently large $r$ .

3.1F5step 2.1algebra∎

By [F5], $T$ is nondecreasing; by step 2.1 the logarithmic ratio in the statement is defined and nonnegative for all sufficiently large $r$ . Its limsup and liminf therefore lie in $[0, \infty]$ .

Depends on

Used by

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Sources

Goldberg–Ostrovskii, Value Distribution of Meromorphic Functions, Ch. 2 §1, definition of order and lower order (standard reference, not scraped)