§14.5.1 Deficiency Relation

14.5.1 Deficiency Relation

One of the reformulations of the Second Fundamental Theorem is characterized in a way so that it comprises the statements of Picard-like theorems in a compact manner (of course, the conclusions are substantially more far-reaching than just the Picard theorems themselves). We first introduce the relevant terminology.

Definition 14.5.1.1.

Let $𝑎 \in \hat{ℂ}$ , $𝑓 : 𝐷 (0, 𝑅) \to \hat{ℂ}$ be meromorphic. For $𝑘 \in ℕ$ , $𝑟 < 𝑅$ , define $𝑛^{(𝑘)} (𝑟, 𝑎, 𝑓)$ be the number of zeros of $𝑓 - 𝑎$ or $1 / 𝑓$ (if $𝑎 = \infty$ ) within $𝐷 (0, 𝑟)$ , where a repeated zero is counted at most $𝑘$ times.

Then define the truncated Nevanlinna counting function (of order $𝑘$ ) by

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

Denote the functions for when $𝑘 = 1$ by $\overline{𝑁} (𝑟, 𝑎, 𝑓) = 𝑁^{(1)} (𝑟, 𝑎, 𝑓)$ and $\overline{𝑛} (𝑟, 𝑎, 𝑓) = 𝑛^{(1)} (𝑟, 𝑎, 𝑓)$ .

Definition 14.5.1.2.

For any $𝑎 \in \hat{ℂ}$ and a meromorphic function $𝑓 : 𝐷 (0, 𝑅) \to \hat{ℂ}$ , define the deficiency (or defect) $𝑎$ in $𝑓$ by

𝛿 (𝑎, 𝑓) = \underset{𝑟 \to 𝑅^{-}}{lim inf} \frac{𝑚 (𝑟, 𝑎, 𝑓)}{𝑇 (𝑟, 𝑓)} = 1 - \underset{𝑟 \to 𝑅^{-}}{lim sup} \frac{𝑁 (𝑟, 𝑎, 𝑓)}{𝑇 (𝑟, 𝑓)} .

Then

𝑎

is said to be a deficient value of

𝑓

if its deficiency is nonzero.

Remark.

Because

𝑁 (𝑟, 𝑎, 𝑓) \leq 𝑇 (𝑟, 𝑓) + 𝒪︀ (1)

holds by the First Fundamental Theorem, the deficiency will always be a nonnegative real value.

The deficiency is the primary value we are interested in. We also observe the use of the Verzweigungsindex (ramification index) $𝜃$ and the truncated deficiency $Θ$ (of order 1):

𝜃 (𝑎, 𝑓) = \underset{𝑟 \to 𝑅^{-}}{lim inf} \frac{𝑁 (𝑟, 𝑎, 𝑓) - \overline{𝑁} (𝑟, 𝑎, 𝑓)}{𝑇 (𝑟, 𝑓)}, Θ (𝑎, 𝑓) = 1 - \underset{𝑟 \to 𝑅^{-}}{lim sup} \frac{\overline{𝑁} (𝑟, 𝑎, 𝑓)}{𝑇 (𝑟, 𝑓)} .

Using these formulations, we now have

Theorem 14.5.1.3 (Deficiency Relation).

Suppose $𝑓 : 𝐷 (0, 𝑅_{0}) \to \hat{ℂ}$ ( $0 < 𝑅_{0} \leq \infty$ ) is meromorphic and admissible. Then $𝑆 = {𝑎 \in \hat{ℂ} : Θ (𝑎, 𝑓) > 0}$ is a countable set and

\sum_{𝑎 \in 𝑆} [𝛿 (𝑎, 𝑓) + 𝜃 (𝑎, 𝑓)] \leq \sum_{𝑎 \in 𝑆} Θ (𝑎, 𝑓) \leq 2 .

Proof.

For any $𝑞$ numbers ${𝑎_{𝜈}}_{𝜈 = 1}^{𝑞} \subset ℂ$ ,

\frac{𝑚 (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} + \sum_{𝜈 = 1}^{𝑞} \frac{𝑚 (𝑟, 𝑎_{𝜈}, 𝑓)}{𝑇 (𝑟, 𝑓)} + \frac{𝑁_{1} (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} \leq 2 + \frac{𝑆 (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)}

as $𝑟 \to 𝑅_{0}^{-}$ by the Second Fundamental Theorem of Nevanlinna Theory (Theorem 14.4.1 $Theorem 14.4.1 Theorem 14.4.1$ ). Then,

\begin{aligned} 𝑞 - 1 - \frac{𝑆 (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} & \leq \frac{𝑁 (𝑟, 𝑓^{'}) - 𝑁 (𝑟, \frac{1}{𝑓^{'}}) - 2 𝑁 (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} \\ \begin{matrix}  \end{matrix} + \frac{𝑁 (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} + \sum_{𝜈 = 1}^{𝑞} \frac{𝑁 (𝑟, 𝑎_{𝜈}, 𝑓)}{𝑇 (𝑟, 𝑓)} . \end{aligned}

At each pole of $𝑓$ with multiplicity $𝑝$ , $𝑓^{'}$ will have a pole of multiplicity $𝑝 + 1$ . Then distinct poles can be counted as $𝑁 (𝑟, 𝑓^{'}) - 𝑁 (𝑟, 𝑓) = \overline{𝑁} (𝑟, 𝑓)$ . Then

𝑞 - 1 - \frac{𝑆 (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} \leq \frac{\overline{𝑁} (𝑟, 𝑓) - 𝑁 (𝑟, 0, 𝑓^{'})}{𝑇 (𝑟, 𝑓)} + \sum_{𝜈 = 1}^{𝑞} \frac{𝑁 (𝑟, 𝑎_{𝜈}, 𝑓)}{𝑇 (𝑟, 𝑓)} .

Each zero $𝑧$ of multiplicity $𝑚$ of $𝑓 - 𝑎_{𝜈}$ gives rise to a zero of multiplicity $𝑚 - 1$ of $𝑓^{'}$ . Then $\overline{𝑛} (𝑟, 𝑎_{𝜈}, 𝑓) = 𝑛 (𝑟, 𝑎_{𝜈}, 𝑓) - 𝑛_{𝑎_{𝜈}} (𝑟, 0, 𝑓^{'})$ where $𝑛_{𝑎_{𝜈}} (𝑟, 0, 𝑓^{'})$ is the zero discrete counting function of $𝑓^{'}$ with the proviso that points are counted at points where $𝑓 = 𝑎_{𝜈}$ . In effect, $𝑁_{𝑎_{𝜈}} (𝑟, 0, 𝑓^{'})$ counts the solutions to $𝑓 = 𝑎_{𝜈}$ , where each solution is under-counted once. Hence, this difference counts these under-counts, or effectively, the number of solutions without consideration of multiplicity.

Then

\sum_{𝜈} \overline{𝑁} (𝑟, 𝑎_{𝜈}, 𝑓) = \sum_{𝜈} 𝑁 (𝑟, 𝑎_{𝜈}, 𝑓) - 𝑁_{{𝑎_{𝜈}}_{𝜈}} (𝑟, 0, 𝑓^{'})

(where $𝑁_{{𝑎_{𝜈}}_{𝜈}} (𝑟, 0, 𝑓^{'})$ is the zero counting function of $𝑓^{'}$ with the condition that the points $𝑧$ satisfy $𝑓 (𝑧) \in {𝑎_{𝜈}}_{𝜈}$ ), and letting $𝑁_{0} (𝑟, 0, 𝑓^{'})$ count the critical points of $𝑓$ that satisfy $𝑓 \notin {𝑎_{𝜈}}_{𝜈}$ ,

𝑁_{{𝑎_{𝜈}}_{𝜈}} (𝑟, 0, 𝑓^{'}) + 𝑁_{0} (𝑟, 0, 𝑓^{'}) = 𝑁 (𝑟, 0, 𝑓^{'}) .

Hence,

𝑞 - 1 - \frac{𝑆 (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} \leq \frac{\overline{𝑁} (𝑟, 𝑓)}{𝑇 (𝑟, 𝑓)} - \frac{𝑁_{0} (𝑟, 0, 𝑓^{'})}{𝑇 (𝑟, 𝑓)} + \sum_{𝜈 = 1}^{𝑞} \frac{\overline{𝑁} (𝑟, 𝑎_{𝜈}, 𝑓)}{𝑇 (𝑟, 𝑓)} . (14.5.1 . 1)

Then taking the superior limits on both side through a sequence ${𝑟_{𝑛}}$ such that ${lim}_{𝑛 \to \infty} \frac{𝑆 (𝑟_{𝑛}, 𝑓)}{𝑇 (𝑟_{𝑛}, 𝑓)} = 0$ (whose existence is given by Theorem 14.4.1.8 $Theorem 14.4.1.8 Theorem 14.4.1.8$ ) and loosening the inequality by removing the $𝑁_{0}$ term,

\begin{matrix} 𝑞 - 1 \leq \underset{𝑛 \to \infty}{lim sup} \frac{\overline{𝑁} (𝑟_{𝑛}, \infty, 𝑓)}{𝑇 (𝑟_{𝑛}, 𝑓)} + \sum_{𝜈 = 1}^{𝑞} \underset{𝑛 \to \infty}{lim sup} \frac{\overline{𝑁} (𝑟_{𝑛}, 𝑎_{𝜈}, 𝑓)}{𝑇 (𝑟_{𝑛}, 𝑓)} \\ \leq \underset{𝑟 \to 𝑅_{0}^{-}}{lim sup} \frac{\overline{𝑁} (𝑟, \infty, 𝑓)}{𝑇 (𝑟, 𝑓)} + \sum_{𝜈 = 1}^{𝑞} \underset{𝑟 \to 𝑅_{0}^{-}}{lim sup} \frac{\overline{𝑁} (𝑟, 𝑎_{𝜈}, 𝑓)}{𝑇 (𝑟, 𝑓)} \\ = 1 - Θ (\infty, 𝑓) + \sum_{𝜈 = 1}^{𝑞} [1 - Θ (𝑎_{𝜈}, 𝑓)] . \end{matrix}

Therefore, $\sum_{𝜈 = 1}^{𝑞} Θ (𝑎_{𝜈}, 𝑓) + Θ (\infty, 𝑓) \leq 2$ . The $Θ (\infty, 𝑓)$ term may or may not be truncated. Without loss of generality, we allow the arbitrary distinct ${𝑎_{𝜈}}_{𝜈}^{𝑞}$ to comprise $\infty$ , to obtain a more uniformized expression.

Assume some arbitrary $𝑁 \in ℕ$ . If $2 𝑁 + 1$ values in the Riemann sphere have truncated deficiencies each exceeding $1 / 𝑁$ , then applying the previous to result to these $2 𝑁 + 1$ values, their truncated deficiency sum exceeds 2, which is a contradiction. Thus, $\forall 𝑁 \in ℕ$ , at most $2 𝑁$ values have truncated deficiencies exceeding $1 / 𝑁$ . Let these exceeding values form the finite set $𝑆_{𝑁}$ . Then

𝑆 = ⋃_{𝑁 = 1}^{\infty} 𝑆_{𝑁} = {𝑎 \in \hat{ℂ} : Θ (𝑎, 𝑓) > 0}

is a countable set, and hence can be described with a sequence ${𝑎_{𝜈}}_{𝜈 = 1}^{𝑞}$ . Then by the inequality,

\sum_{𝜈 = 1}^{𝑞} Θ (𝑎_{𝜈}, 𝑓) \leq 2 ⟹ \sum_{𝜈 = 1}^{\infty} Θ (𝑎_{𝜈}, 𝑓) \leq 2 . □

Notice that the method we used to prove countability is the exact same as in Corollary 10.2.3.1 $Corollary 10.2.3.1 Corollary 10.2.3.1$ .

Remark.

Ahlfors’ geometric formulation of the second fundamental theorem suggests that the value 2 in the inequality is connected to the Euler characteristic of the Riemann sphere via triangulation.

Therefore, we have

Corollary 14.5.1.3.1.

A non-constant meromorphic function omits at most two values on the Riemann sphere.

Proof.

Assume there exists three values $𝑎$ , $𝑏$ , $𝑐 \in \hat{ℂ}$ that meromorphic function $𝑓$ does not ever attain. Then $𝑁 (𝑟, 𝑎, 𝑓)$ , $𝑁 (𝑟, 𝑏, 𝑓)$ , $𝑁 (𝑟, 𝑐, 𝑓)$ all vanish identically with respect to radius while $𝑇 > 0$ .

Then $𝛿 (𝑎, 𝑓)$ , $𝛿 (𝑏, 𝑓)$ , $𝛿 (𝑐, 𝑓) = 1$ . By the deficiency relation (Theorem 14.5.1.3 $Theorem 14.5.1.3 Theorem 14.5.1.3$ ),

2 \geq \sum_{𝑧 \in 𝑆} [𝛿 (𝑧) + 𝜃 (𝑧)] \geq 𝛿 (𝑎, 𝑓) + 𝛿 (𝑏, 𝑓) + 𝛿 (𝑐, 𝑓) = 3,

which is an impossibility. ⁠ $□$

Corollary 14.5.1.3.2 (Picard’s Little Theorem).

Any non-constant entire function omits at most one value.

Proof.

Any such function can be regarded as a meromorphic function on the plane that omits

\infty

. Then the conclusion follows from Corollary 14.5.1.3.1

Corollary 14.5.1.3.1 Corollary 14.5.1.3.1

. ⁠

□

The use of spherically normal families in the proof of the Great Picard Theorem for more general domains is unavoidable. For further treatment, refer to [5].