§14.5.2 Functions Sharing Values

14.5.2 Functions Sharing Values

In Nevanlinna’s explanation of the various applications of the Second Fundamental Theorem, he proved:

Theorem 14.5.2.1 (Nevanlinna’s Five-Value Theorem).

Let $𝑓_{1}, 𝑓_{2}$ be two meromorphic, admissible functions on $𝐷 (0, 𝑅_{0})$ ( $𝑅_{0} < \infty$ ). For each $𝑎 \in \hat{ℂ}$ , let $𝐸_{1} (𝑎) = {𝑧 : 𝑓_{1} (𝑧) = 𝑎}$ , $𝐸_{2} (𝑎) = {𝑧 : 𝑓_{2} (𝑧) = 𝑎}$ . Then if $𝐸_{1} (𝑎) \equiv 𝐸_{2} (𝑎)$ for at least five different values¹ of $𝑎 \in \hat{ℂ}$ , then $𝑓_{1} \equiv 𝑓_{2}$ .

If $𝑓_{1}, 𝑓_{2}$ are two merely meromorphic (not necessarily admissible) functions on $ℂ$ and if $𝐸_{1} (𝑎) \equiv 𝐸_{2} (𝑎)$ for at least five different values of $𝑎 \in \hat{ℂ}$ , then either $𝑓_{1} \equiv 𝑓_{2}$ everywhere or they are both constant.

Proof.

Assume that five values $𝑎_{1}$ to $𝑎_{5}$ exist for which $𝐸_{1} (𝑎_{1}) = 𝐸_{2} (𝑎_{1}), \dots, 𝐸_{1} (𝑎_{5}) = 𝐸_{2} (𝑎_{5})$ . Without loss of generality, assume they are finite (otherwise a Möbius transformation can be utilized to map a value outside these five to the infinity point).

Because

\overline{𝑁} (𝑟, 𝑎_{𝜈}, 𝑓_{1}) = \overline{𝑁} (𝑟, 𝑎_{𝜈}, 𝑓_{2})

for $𝜈 = 1, \dots, 5$ , $0 < 𝑟 < 𝑅_{0}$ , quantify this value by $𝑁_{𝜈} (𝑟)$ . We now assume that $𝑓_{1} \notin \equiv 𝑓_{2}$ . Now each shared value corresponds to an infinity of $Ψ = 1 / (𝑓_{2} - 𝑓_{1})$ .

Now assuming that $𝑓_{1}$ , $𝑓_{2}$ are not identically equal, we have

\begin{aligned} \sum_{𝜈 = 1}^{5} 𝑁_{𝜈} (𝑟) & \leq 𝑁 (𝑟, Ψ) \leq 𝑇 (𝑟, Ψ) = 𝑇 (𝑟, 𝑓_{2} - 𝑓_{1}) + 𝒪︀ (1) \\ \leq 𝑇 (𝑟, 𝑓_{1}) + 𝑇 (𝑟, 𝑓_{2}) + 𝒪︀ (1) \end{aligned}

trivially by the First Fundamental Theorem (Theorem 14.2.4 $Theorem 14.2.4 Theorem 14.2.4$ ) and the properties of the characteristic. Applying (14.5.1.1) $(14.5.1.1) (14.5.1.1)$ gives

\begin{aligned} 4 𝑇 (𝑟, 𝑓_{𝑗}) - 𝑆 (𝑟, 𝑓_{𝑗}) & \leq \overline{𝑁} (𝑟, 𝑓_{𝑗}) + \sum_{𝜈 = 1}^{5} 𝑁_{𝜈} (𝑟) - 𝑁_{0} (𝑟, 0, 𝑓_{𝑗^{'}}) 𝑗 = 1, 2, \\ 3 𝑇 (𝑟, 𝑓_{𝑗}) & \leq 𝑆 (𝑟, 𝑓_{𝑗}) + \sum_{𝜈 = 1}^{5} 𝑁_{𝜈} (𝑟) . \end{aligned}

Summing this inequality for both values of $𝑗$ and applying the upper bound gives

\begin{aligned} 𝑇 (𝑟, 𝑓_{1}) + 𝑇 (𝑟, 𝑓_{2}) & \leq \frac{1}{3} [𝑆 (𝑟, 𝑓_{1}) + 𝑆 (𝑟, 𝑓_{2})] + \frac{2}{3} \sum_{𝜈 = 1}^{5} 𝑁_{𝜈} (𝑟) \\ \leq \frac{1}{3} [𝑆 (𝑟, 𝑓_{1}) + 𝑆 (𝑟, 𝑓_{2})] + \frac{2}{3} [𝑇 (𝑟, 𝑓_{1}) + 𝑇 (𝑟, 𝑓_{2})] + 𝒪︀ (1) . \end{aligned}

If $𝑅_{0} < \infty$ , then outside a set $𝐸_{0}$ of logarithmic measure not exceeding $4$ , or $\int_{𝐸_{0}} \frac{d 𝑟}{𝑅_{0} - 𝑟} \leq 4$ , by the Second Fundamental Theorem estimates (Theorem 14.4.1.5 $Theorem 14.4.1.5 Theorem 14.4.1.5$ ) and the fact that ${log}^{+} 𝑇 (𝑟, 𝑓) = ℴ︀ (𝑇 (𝑟, 𝑓))$ ,

\begin{aligned} (1 - ℴ︀ (1)) [𝑇 (𝑟, 𝑓_{1}) + 𝑇 (𝑟, 𝑓_{2})] & \leq 𝒪︀ (log \frac{1}{𝑅_{0} - 𝑟}) \\ ⟹ \underset{\begin{matrix} 𝑟 \to 𝑅_{0}^{-} \\ 𝑟 \notin 𝐸_{0} \end{matrix}}{lim sup} \frac{𝑇 (𝑟, 𝑓_{𝑗})}{log \frac{1}{𝑅_{0} - 𝑟}} = 𝒪︀ (1) . \end{aligned}

We now aim to extend this outside of $𝐸_{0}$ .

For any $𝑟^{*} \in 𝐸_{0}$ , let $𝑟^{'} \in {𝑟_{𝑛}}$ be the infimum of values exceeding $𝑟^{*}$ not in $𝐸_{0}$ . If $𝑟^{*} = 𝑟^{'}$ , then arbitrarily close to $𝑟^{*}$ , there are points for which the limit superior assertion holds. By continuity, this holds at $𝑟^{*}$ as well.

Hence, assume $𝑟^{'} > 𝑟^{*}$ . Then $[𝑟^{*}, 𝑟_{𝑛}) \subseteq 𝐸_{0}$ , we must have $\int_{𝑟^{*}}^{𝑟_{𝑛}} \frac{d 𝑟}{𝑅_{0} - 𝑟} = log (\frac{𝑅_{0} - 𝑟^{*}}{𝑅_{0} - 𝑟^{'}}) \leq 4$ , so $log (\frac{1}{𝑅_{0} - 𝑟^{*}}) \geq log (\frac{1}{𝑅_{0} - 𝑟^{'}}) - 4$ . Since $log (\frac{1}{𝑅_{0} - 𝑟})$ is continuous and non-vanishing for $𝑟$ sufficiently close to $𝑅_{0}$ and $𝑇$ is non-decreasing,

\frac{𝑇 (𝑟^{*}, 𝑓_{𝑗})}{log \frac{1}{𝑅_{0} - 𝑟^{*}}} \leq \frac{𝑇 (𝑟^{'}, 𝑓_{𝑗})}{log \frac{1}{𝑅_{0} - 𝑟^{'}}} (1 + \frac{4}{log \frac{1}{𝑅_{0} - 𝑟^{'}} - 4}) = 𝒪︀ (\frac{𝑇 (𝑟^{'}, 𝑓_{𝑗})}{log \frac{1}{𝑅_{0} - 𝑟^{'}}}) .

This shows that

\underset{𝑟 \to 𝑅_{0}^{-}}{lim sup} \frac{𝑇 (𝑟, 𝑓_{𝑗})}{log \frac{1}{𝑅_{0} - 𝑟}} = 𝒪︀ (1),

which contradicts the admissibility of $𝑓_{𝑗}$ . Hence, our original assumption that $𝑓_{1} ≢ 𝑓_{2}$ is false, completing the first statement of the theorem.

On the other hand, if $𝑅_{0} = \infty$ , then outside a set $𝐸_{0}$ of linear measure not exceeding $4$ , by the Second Fundamental Theorem estimates (Theorem 14.4.1.5 $Theorem 14.4.1.5 Theorem 14.4.1.5$ ), $𝑆 (𝑟, 𝑓_{𝑗}) = ℴ︀ (𝑇 (𝑟, 𝑓_{𝑗}))$ , and thus

\begin{aligned} 𝑇 (𝑟, 𝑓_{1}) + 𝑇 (𝑟, 𝑓_{2}) & \leq ℴ︀ (𝑇 (𝑟, 𝑓_{1})) + ℴ︀ (𝑇 (𝑟, 𝑓_{2})) + 𝒪︀ (1) \\ ⟹ 𝑇 (𝑟, 𝑓_{1}) + 𝑇 (𝑟, 𝑓_{2}) = 𝒪︀ (1) . \end{aligned}

Then $𝑓_{1}, 𝑓_{2}$ are constant, or the assumption that $𝑓_{1} ≢ 𝑓_{2}$ was false. This concludes the theorem. ⁠ $□$

Remark.

The functions $𝑓_{1} : 𝑧 \mapsto exp (𝑧)$ and $𝑓_{2} : 𝑧 \mapsto exp (- 𝑧)$ imply that five cannot be sharpened to four. Indeed,

𝐸_{1} (0) = 𝐸_{2} (0) = ⌀ = 𝐸_{1} (\infty) = 𝐸_{2} (\infty),

𝐸_{1} (1) = 𝐸_{2} (1) = {2 π i 𝑘}_{𝑘 \in ℤ},

and

𝐸_{1} (- 1) = 𝐸_{2} (- 1) = {2 π i 𝑘 + π i}_{𝑘 \in ℤ} .

It can be verified algebraically that

0

\infty

1

- 1

are the only “shared” values; the first two are omitted values while any solution must satisfy

e^{2 𝑧} = 1

, which corresponds to points where

exp

attains

1

and

- 1

The recent developments of Nevanlinna has generalized preexisting results to more arbitrary domains such as annuli. A generalization of the Five-Value theorem on annuli is given in [6].

Further results on Nevanlinna theory, such as the value distribution of derivatives, the Ahlfors’ theory, and the proofs of the theorems given at the end of Section 8 $Section 8 Section 8$ , we refer to [5].