§8.3 From Schwarz–Pick to Ahlfors and Value Distribution of Entire Functions

8.3 From Schwarz–Pick to Ahlfors and Value Distribution of Entire Functions

While Schwarz Lemma in Lemma 3.5.1 $Lemma 3.5.1 Lemma 3.5.1$ concerns self-maps of $𝔻$ with a fixed point at the origin, the Schwarz–Pick Lemma in Lemma 3.5.4 $Lemma 3.5.4 Lemma 3.5.4$ generalizes this to arbitrary points in $𝔻$ as well as the hyperbolic contraction property of holomorphic maps.

In 1938, Lars Ahlfors provided a further generalization by curvature, prompting the study of complex functions from a differential-geometric approach.

The hyperbolic metric $𝜆$ in (8.2.2) $(8.2.2) (8.2.2)$ does not increase under any holomorphic $𝑓 : 𝔻 \to 𝔻$ . It was realized that this was a consequence of the constant negative curvature $- 1$ of $𝜆$ . The results we now provide are simplifications of those from [2].

Theorem 8.3.1 (Schwarz–Ahlfors–Pick).

Let $𝑓$ be holomorphic on $𝔻$ . Suppose that $𝜌$ is a regular metric defined on an open neighborhood $𝑈$ , where $𝑓 (𝔻) \subseteq 𝑈$ , $d 𝑠_{𝜌}^{2} = 𝜌^{2} (𝑤) {| d 𝑤 |}^{2}$ , and $𝐾_{𝜌} (𝑤) \leq - 1$ for all $𝑤 \in 𝑈$ . Then

𝑓^{*} 𝜌 (𝑧) \leq 𝜆 (𝑧) \forall 𝑧 \in 𝔻,

where $𝜆$ is the Poincaré metric, and equivalently,

d 𝑠_{𝑓^{*} 𝜌}^{2} \leq d 𝑠_{𝜆}^{2},

or that the metric

𝜌

does not exceed the hyperbolic metric under the map

𝑓

Proof.

Define

𝜆_{𝑟} (𝑧) = {(𝑧 \mapsto \frac{𝑧}{𝑟})}^{*} 𝜆 (𝑧) = \frac{2 𝑟}{𝑟^{2} - {| 𝑧 |}^{2}}, 0 < 𝑟 < 1 (8.3 . 1)

to generalize the Poincaré metric to $𝐷 (0, 𝑟)$ . (8.2.5) $(8.2.5) (8.2.5)$ gives that $𝐾_{𝜆_{𝑟}} (𝑧) = 𝐾_{𝜆} (\frac{𝑧}{𝑟}) = - 1$ for any $𝑧 \in 𝐷 (0, 𝑟)$ . Define the real-valued function

𝑢_{𝑟} (𝑧) = \frac{𝑓^{*} 𝜌 (𝑧)}{𝜆_{𝑟} (𝑧)} for 𝑧 \in 𝐷 (0, 𝑟),

which is nonnegative and continuous on $𝐷 (0, 𝑟)$ . The pullback metric $𝑓^{*} 𝜌 = (𝜌 \circ 𝑓) | 𝑓^{'} |$ is continuous on $𝔻$ and thus bounded on $\overline{𝐷 (0, 𝑟)}$ (as a consequence of Theorem 1.2.22 $Theorem 1.2.22 Theorem 1.2.22$ ). As $| 𝑧 | \to 𝑟^{-}$ , $𝜆_{𝑟} (𝑧) \to \infty$ , and hence ${lim}_{| 𝑧 | \to 𝑟^{-}} 𝑢_{𝑟} (𝑧) = 0$ . Thus,

𝑀_{𝑟} = max_{𝑧 \in \overline{𝐷 (0, 𝑟)}} 𝑢_{𝑟} (𝑧)

must be attained at some $𝑧 = 𝜏_{𝑟} \in 𝐷 (0, 𝑟)$ (within the interior).

If $𝑀_{𝑟} = 0$ , then $\forall 𝑧 \in 𝐷 (0, 𝑟)$ , $\frac{𝑓^{*} 𝜌 (𝑧)}{𝜆_{𝑟} (𝑧)} = 0 ⟹ 𝑀_{𝑟} = 0 \leq 1$ by maximality. On the contrary, if $𝑀_{𝑟} > 0$ , $𝑓^{*} 𝜌$ has well-defined Gaussian curvature at $𝜏_{𝑟}$ . Since

\begin{aligned} (\nabla^{2} log (𝑢_{𝑟})) (𝜏_{𝑟}) & = (\nabla^{2} log (𝑓^{*} 𝜌)) (𝜏_{𝑟}) - (\nabla^{2} log (𝜆_{𝑟})) (𝜏_{𝑟}) \\ = - 𝐾_{𝑓^{*} 𝜌} (𝜏_{𝑟}) 𝑓^{*} {𝜌 (𝜏_{𝑟})}^{2} + 𝐾_{𝜆_{𝑟}} (𝜏_{𝑟}) 𝜆_{𝑟} {(𝜏_{𝑟})}^{2} \\ = - 𝐾_{𝑓^{*} 𝜌} (𝜏_{𝑟}) 𝑓^{*} {𝜌 (𝜏_{𝑟})}^{2} - 𝜆_{𝑟} {(𝜏_{𝑟})}^{2} . (8.3 . 2) \end{aligned}

By assumption, we have $- 𝐾_{𝑓^{*} 𝜌} (𝜏_{𝑟}) \geq 1$ . Hence, $(\nabla^{2} log (𝑢_{𝑟})) (𝜏_{𝑟}) \geq 𝑓^{*} {𝜌 (𝜏_{𝑟})}^{2} - 𝜆_{𝑟} {(𝜏_{𝑟})}^{2}$ . Since $log$ is increasing in $ℝ$ , $𝜏_{𝑟}$ is a local maximum of $log \circ 𝑢_{𝑟}$ and hence $(\nabla^{2} log (𝑢_{𝑟})) (𝜏_{𝑟}) \leq 0$ . Thus, we have

𝑓^{*} {𝜌 (𝜏_{𝑟})}^{2} - 𝜆_{𝑟} {(𝜏_{𝑟})}^{2} \leq 0 ⟺ 𝑀_{𝑟} \leq 1 .

Now let $𝑟 \to 1^{-}$ , and it follows that $𝑀_{𝑟} \to {sup}_{𝑧 \in 𝔻} \frac{𝑓^{*} 𝜌 (𝑧)}{𝜆 (𝑧)} \leq 1$ . ⁠ $□$

Theorem 8.3.2.

Let $𝑓 : 𝔻 \to 𝑈 \subseteq ℂ$ be holomorphic. Let $d 𝑠 = 𝜌 (𝑤) | d 𝑤 |$ (where $𝜌 : 𝑈 \to ℝ_{> 0}$ ) define a regular metric such that at every point $𝑤 \in 𝑈$ , either

The second derivatives of $log 𝜆$ are continuous ( $𝐶^{2}$ ) and
$\nabla^{2} (log 𝜆) (𝑤) \geq 𝜆^{2}$
There exist two opposite directions ${\hat{𝐧}}^{'}$ , ${\hat{𝐧}}^{″}$ such that
$\nabla_{{\hat{𝐧}}^{'}} (log 𝜌) (𝑤) + \nabla_{{\hat{𝐧}}^{″}} (log 𝜌) (𝑤) > 0 (8.3 . 3)$
(the directional derivatives).

Then the metric $𝑓^{*} 𝜌$ does not exceed the hyperbolic metric $𝜆$ .

Proof.

The first case is equivalent to $𝐾_{𝜌} \leq - 1$ .

The only modification to the proof of Theorem 8.3.1 $Theorem 8.3.1 Theorem 8.3.1$ is to consider the case of the inequality involving directional derivatives for each $𝜏_{𝑟}$ (which by definition, is where the maximum value of $𝑢_{𝑟}$ is attained within $𝐷 (0, 𝑟)$ ). By the increasing nature of $log$ , $𝜏_{𝑟}$ is also a local maximum of $𝜙 \equiv log 𝑢_{𝑟} \equiv log (𝑓^{*} 𝜌) - log 𝜆_{𝑟}$ .

Since $𝜏_{𝑟}$ is a local maximum, we must have

\nabla_{{\hat{𝐧}}^{'}} 𝜙 (𝜏_{𝑟}) \leq 0, \nabla_{{\hat{𝐧}}^{″}} 𝜙 (𝜏_{𝑟}) \leq 0 ⟹ \nabla_{{\hat{𝐧}}^{'}} 𝜙 (𝜏_{𝑟}) + \nabla_{{\hat{𝐧}}^{″}} 𝜙 (𝜏_{𝑟}) \leq 0 .

This implies that

\begin{aligned} \nabla_{{\hat{𝐧}}^{'}} (log 𝑓^{*} 𝜌) (𝜏_{𝑟}) + \nabla_{{\hat{𝐧}}^{″}} (log 𝑓^{*} 𝜌) (𝜏_{𝑟}) \\ \begin{matrix}  \end{matrix} \leq \nabla_{{\hat{𝐧}}^{'}} (log 𝜆_{𝑟}) (𝜏_{𝑟}) + \nabla_{{\hat{𝐧}}^{″}} (log 𝜆_{𝑟}) (𝜏_{𝑟}) = 0 \end{aligned}

by the symmetry of the hyperbolic metric and the fact that the two directions are opposite to each other. Pulling back to $𝜌$ contradicts with (8.3.3) $(8.3.3) (8.3.3)$ . Thus, $𝜏_{𝑟}$ cannot both simultaneously be the location of a maximum while satisfying said inequality; therefore the theorem follows. ⁠ $□$

Theorem 8.3.3.

Let $𝑓 : 𝔻 \to 𝑈$ be holomorphic. Let

𝜌 : 𝑈 \subseteq ℂ \to ℝ_{> 0}, d 𝑠_{𝜌} = 𝜌 (𝑤) | d 𝑤 |

be a continuous conformal metric (but not necessarily

𝐶^{2}

) such that at each point

𝑤

, there exists a neighborhood

𝑉_{𝑤} ∋ 𝑤

𝑈

and a regular metric

𝜌_{𝑤}

thereon such that

𝜌_{𝑤} (𝑤) = 𝜌 (𝑤)

and

𝜌_{𝑤} \leq 𝜌

everywhere else (referred to as a “supporting metric”). If each

𝐾_{𝜌_{𝑤}} \leq - 1

everywhere, then the conclusion of Theorem 8.3.1

Theorem 8.3.1 Theorem 8.3.1

continues to hold for

𝜌

Proof.

By assumption, we have

𝑓^{*} 𝜌_{𝜏_{𝑟}} (𝜏_{𝑟}) = 𝑓^{*} 𝜌 (𝜏_{𝑟}), 𝑓^{*} 𝜌_{𝜏_{𝑟}} (𝑧) \leq 𝑓^{*} 𝜌 (𝑧) for 𝑧 \neq 𝜏_{𝑟} .

Let ${\tilde{𝑢}}_{𝑟} (𝑧) = \frac{𝑓^{*} 𝜌_{𝜏_{𝑟}} (𝑧)}{𝜆_{𝑟} (𝑧)}$ , which attains is maximum of $𝑀_{𝑟}$ at $𝜏_{𝑟}$ as well.

The calculations in Theorem 8.3.1 $Theorem 8.3.1 Theorem 8.3.1$ on ${\tilde{𝑢}}_{𝑟} (𝑧)$ (whose curvature calculations are now valid by $𝐶^{2}$ ) give that $𝑓^{*} 𝜌_{𝜏_{𝑟}} {(𝜏_{𝑟})}^{2} - 𝜆_{𝑟} {(𝜏_{𝑟})}^{2} \leq 0$ , which implies $𝑀_{𝑟} \leq 1$ . (We have used the supporting metric, rather than $𝜌$ , to derive this inequality) The rest of the theorem follows naturally. ⁠ $□$

Theorem 8.3.1 $Theorem 8.3.1 Theorem 8.3.1$ generalizes the Schwarz–Pick Theorem when $𝜌$ is chosen to be $𝜆$ and $𝑓$ is chosen such that $𝑓 (𝔻) \subseteq 𝔻$ .

For the purpose of the proceeding generalization, we define the conformal metric

𝜆_{𝑟}^{𝛼} (𝑧) = \frac{1}{\sqrt{𝛼}} {(𝑧 \mapsto \frac{𝑧}{𝑟})}^{*} 𝜆 (𝑧) = \frac{2 𝑟}{\sqrt{𝛼} (𝑟^{2} - {| 𝑧 |}^{2})}, 𝑟 > 0, 𝑧 \in 𝐷 (0, 𝑟) . (8.3 . 4)

Its Gaussian curvature is

\begin{aligned} 𝐾_{𝜆_{𝑟}^{𝛼}} (𝑧) & = - 4 \frac{𝜕^{2}}{𝜕 \overline{𝑧} 𝜕 𝑧} [log (\frac{2 𝑟}{\sqrt{𝛼} (𝑟^{2} - {| 𝑧 |}^{2})})] {(\frac{\sqrt{𝛼} (𝑟^{2} - {| 𝑧 |}^{2})}{2 𝑟})}^{2} \\ = - 4 𝛼 \frac{𝜕^{2}}{𝜕 \overline{𝑧} 𝜕 𝑧} [log (\frac{2 𝑟}{𝑟^{2} - {| 𝑧 |}^{2}})] {(\frac{𝑟^{2} - {| 𝑧 |}^{2}}{2 𝑟})}^{2} = 𝛼 𝐾_{𝜆_{𝑟}} (𝑧) = - 𝛼, \end{aligned}

via the results and definitions in (8.3.1) $(8.3.1) (8.3.1)$ .

Corollary 8.3.3.1.

Let $𝑟 > 0$ and suppose $𝑓 : 𝐷 (0, 𝑟) \to 𝑈$ is holomorphic, where $𝑈 \subseteq ℂ$ is a region. For any $𝛽 > 0$ , define $𝜌$ to be a regular metric on $𝑈$ with $d 𝑠_{𝜌}^{2} = 𝜌^{2} (𝑤) {| d 𝑤 |}^{2}$ such that

𝐾_{𝜌} (𝑤) \leq - 𝛽, \forall 𝑤 \in 𝑈 .

Then $\forall 𝛼 > 0$ ,

𝑓^{*} 𝜌 (𝑧) \leq \sqrt{\frac{𝛼}{𝛽}} 𝜆_{𝑟}^{𝛼} (𝑧)

for any

𝑧 \in 𝐷 (0, 𝑟)

, where

𝑓^{*} 𝜌 (𝑧) = (𝜌 \circ 𝑓) | 𝑓^{'} |

is the metric pullback.

Proof.

Consider the ${(𝑧 \mapsto 𝑧 𝑟)}^{*} 𝑓^{*} (𝜌 \sqrt{𝛽})$ , a conformal metric pullback of $𝜌 \sqrt{𝛽}$ to $𝔻$ , which satisfies

𝐾_{{(𝑧 \mapsto 𝑧 𝑟)}^{*} 𝑓^{*} (𝜌 \sqrt{𝛽})} \leq - 1 .

By Schwarz–Ahlfors–Pick (Theorem 8.3.1 $Theorem 8.3.1 Theorem 8.3.1$ ), we have

{(𝑧 \mapsto 𝑧 𝑟)}^{*} 𝑓^{*} (𝜌 \sqrt{𝛽}) (𝑧) \leq 𝜆 (𝑧) = \sqrt{𝛼} ({(𝑧 \mapsto 𝑟 𝑧)}^{*} 𝜆_{𝑟}^{𝛼}) (𝑧) for 𝑧 \in 𝔻 .

Since $𝑟 \neq 0$ , this implies that

𝑓^{*} (𝜌 \sqrt{𝛽}) (𝑧) \leq \sqrt{𝛼} 𝜆_{𝑟}^{𝛼} (𝑧) for 𝑧 \in 𝐷 (0, 𝑟) .

Since $\sqrt{𝛽}$ is a constant,

\sqrt{𝛽} 𝑓^{*} 𝜌 (𝑧) \leq \sqrt{𝛼} 𝜆_{𝑟}^{𝛼} (𝑧), \forall 𝑧 \in 𝐷 (0, 𝑟) . □

Corollary 8.3.3.2 (Generalized Liouville).

𝑓 : ℂ \to 𝑈

is entire and

𝑈

admits a regular metric of curvature bounded above by a negative constant, then

𝑓

must be constant.

Proof.

By assumption, $\exists 𝛽 > 0$ such that ${sup}_{𝑤 \in 𝑈} 𝐾_{𝜌} (𝑤) \leq - 𝛽$ . Then Corollary 8.3.3.1 $Corollary 8.3.3.1 Corollary 8.3.3.1$ gives that

𝑓^{*} 𝜌 (𝑧) \leq \frac{1}{\sqrt{𝛽}} 𝜆_{𝑟} (𝑧) \forall 𝑧 \in 𝐷 (0, 𝑟)

for any

𝑟 > 0

. As

𝑟 \to \infty

𝜆_{𝑟} \to 0

. Hence,

𝑓^{*} 𝜌 (𝑧) = 0

, implying that

(𝜌 \circ 𝑓) (𝑧) | 𝑓^{'} | = 0

. Hence,

𝑓

is constant. ⁠

□

Remark.

Corollary 8.3.3.2

Corollary 8.3.3.2 Corollary 8.3.3.2

implies Liouville’s Theorem (Theorem 3.2.5

Theorem 3.2.5 Theorem 3.2.5

). To justify this differential-geometric generalization, suppose

𝑓 : ℂ \to 𝑈

is entire such that

𝑈

is bounded. There then exists some

𝑅 > 0

such that

𝑈 \subseteq 𝐷 (0, 𝑅)

. The metric

𝜆_{𝑅}

has constant negative curvature

𝐾 = - 1

𝐷 (0, 𝑅)

, and hence, under

𝛽 = 1

, Corollary 8.3.3.2

Corollary 8.3.3.2 Corollary 8.3.3.2

implies that

𝑓

is constant.

It is understood that an entire function is guaranteed to be constant if it is bounded. This is a statement of sufficiency, but it begs the question of the capacity for possible generalization of boundedness under which constancy is still always satisfied.

Consider an entire function $𝑓 : ℂ \to 𝑈$ , where $𝑈$ is an unbounded region such that $ℂ ∖ 𝑈$ has positive area. Fix $𝜁 \in int (ℂ ∖ 𝑈)$ . Then the map $𝑧 \mapsto \frac{1}{𝑧 - 𝜁}$ maps $𝑈$ to a bounded region and hence $𝑧 \mapsto \frac{1}{𝑓 (𝑧) - 𝜁}$ is constant by Liouville’s Theorem (Theorem 3.2.5 $Theorem 3.2.5 Theorem 3.2.5$ ), implying the constancy of $𝑓$ (the essential proof of Theorem 4.2.5 $Theorem 4.2.5 Theorem 4.2.5$ ).

In contrast, if $𝑓 : ℂ \to 𝑈$ is entire and $ℂ ∖ 𝑈$ has zero area (one readily considers sets consisting of curves or isolated points), we must be more specific in determining sufficient conditions that still imply constancy of $𝑓$ .

Similar to in the proof of the Riemann Mapping Theorem (Theorem 5.3.2 $Theorem 5.3.2 Theorem 5.3.2$ ), one may use holomorphic square roots or other transformations to reduce to the bounded setting.

Proposition 8.3.4.

𝑓 : ℂ \to ℂ ∖ {𝑥 \in ℝ : 0 \leq 𝑥 \leq 1}

is entire, then

𝑓

must be constant.

Proof.

Consider the biholomorphism $𝜑 (𝑧) = \frac{1}{𝑧}$ , mapping $ℂ ∖ {𝑥 \in ℝ : 0 \leq 𝑥 \leq 1}$ to $ℂ^{*} ∖ ℝ_{\geq 1}$ . By simple connectivity of $ℂ ∖ ℝ_{\geq 1}$ , there exists a univalent branch $𝜓$ of $𝑧 \mapsto \sqrt{𝑧 - 1}$ on $ℂ ∖ ℝ_{\geq 1}$ . Now omitting the origin, it is trivially realized that $𝜓 (ℂ^{*} ∖ ℝ_{\geq 1}) \cap - 𝜓 (ℂ^{*} ∖ ℝ_{\geq 1}) = ⌀$ . If otherwise, then $\exists 𝜉 \in 𝜓 (ℂ^{*} ∖ ℝ_{\geq 1})$ such that $- 𝜉 \in 𝜓 (ℂ^{*} ∖ ℝ_{\geq 1})$ , implying that $\exists 𝑧_{1}, 𝑧_{2} \in ℂ^{*} ∖ ℝ_{\geq 1}$ such that $𝜙 (𝑧_{1}) = 𝜉$ and $𝜙 (𝑧_{2}) = - 𝜉$ , implying that $𝑧_{1} = 𝑧_{2}$ and $𝜉 = 0 ⟹ 𝑧_{1} = 𝑧_{2} = 1$ , which does not lie in $𝜓 (ℂ^{*} ∖ ℝ_{\geq 1})$ .

Now fix $𝜉 \in 𝜓 (ℂ^{*} ∖ ℝ_{\geq 1})$ . By the Open Mapping Theorem (Theorem 5.1.1 $Theorem 5.1.1 Theorem 5.1.1$ ), $\exists 𝜀 > 0$ such that $𝐷 (𝜉, 𝜀) \subseteq 𝜓 (ℂ^{*} ∖ ℝ_{\geq 1})$ . Consequently, $𝐷 (- 𝜉, 𝜀) \cap 𝜓 (ℂ^{*} ∖ ℝ_{\geq 1}) = ⌀$ . Lastly, the function $𝜙 (𝑧) = \frac{𝜀}{𝑧 + 𝜉}$ maps $𝜓 (ℂ^{*} ∖ ℝ_{\geq 1})$ to $𝔻$ . By Liouville (Theorem 3.2.5 $Theorem 3.2.5 Theorem 3.2.5$ ), $𝜙 \circ 𝜓 \circ 𝜑 \circ 𝑓$ is constant, which implies $𝑓$ is constant by the injectivity of $𝜙$ , $𝜓$ , and $𝜑$ . ⁠ $□$

The preceding examples show that if the omitted set is sufficiently “large” (in the sense of having positive area or disconnecting the plane in certain ways), then any entire function avoiding it must reduce to a constant. However, there are natural limits to the smallness of the omitted set. For instance, the exponential function $exp$ is an entire non-constant function whose image is $ℂ^{*}$ , omitting only a single point. Thus, the property that an entire function omits a set is not by itself sufficient to guarantee constancy unless that set is suitably substantial. This observation is formalized by Picard’s Little Theorem (Theorem 8.3.6 $Theorem 8.3.6 Theorem 8.3.6$ ), which as preluded to before, asserts that any non-constant entire function can omit at most one complex value.

Proposition 8.3.5.

Let $𝑈 \subset ℂ$ be an open set such that $ℂ ∖ 𝑈$ contains at least two points. Then $𝑈$ admits a conformal metric $𝜌 \in 𝐶^{2} (𝑈)$ , $d 𝑠_{𝜌}^{2} = 𝜌^{2} (𝑧) {| d 𝑧 |}^{2}$ such that

𝐾_{𝜌} (𝑧) \leq - 𝛽 < 0 \forall 𝑧 \in 𝑈

for some

𝛽 > 0

Proof.

Without loss of generality, we may assume that ${0, 1} \subseteq ℂ ∖ 𝑈$ (if not, an affine complex transformation $𝑧 \mapsto \frac{𝑧 - 𝜉_{1}}{𝜉_{2} - 𝜉_{1}}$ where $𝜉_{1}, 𝜉_{2} \in ℂ ∖ 𝑈$ are distinct will suffice to transform $𝑈$ to such a region).

Define a regular metric with

𝜌 (𝑧) = \frac{\sqrt{1 + {| 𝑧 |}^{\frac{1}{3}}} \sqrt{1 + {| 𝑧 - 1 |}^{\frac{1}{3}}}}{{| 𝑧 |}^{\frac{5}{6}} {| 𝑧 - 1 |}^{\frac{5}{6}}}, d 𝑠_{𝜌}^{2} = {𝜌 (𝑧)}^{2} {| d 𝑧 |}^{2} (8.3 . 5)

on $ℂ ∖ {0, 1}$ .

Since $\nabla^{2} (log {| 𝑧 |}^{\frac{5}{6}}) = \frac{5}{6} \nabla^{2} (log | 𝑧 |) = \frac{5}{6} \nabla^{2} (ℜ𝔢 log (𝑧)) = 0$ by harmonicity,

\begin{aligned} \nabla^{2} log (\frac{\sqrt{1 + {| 𝑧 |}^{\frac{1}{3}}}}{{| 𝑧 |}^{\frac{5}{6}}}) & = 2 \frac{𝜕^{2}}{𝜕 \overline{𝑧} 𝜕 𝑧} (log (1 + {| 𝑧 |}^{\frac{1}{3}})) = \frac{𝑧^{- \frac{5}{6}}}{3} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{{\overline{𝑧}}^{\frac{1}{6}}}{1 + {| 𝑧 |}^{\frac{1}{3}}}) \\ = \frac{𝑧^{- \frac{5}{6}}}{3} \frac{𝜕}{𝜕 \overline{𝑧}} (\frac{{\overline{𝑧}}^{\frac{1}{6}}}{1 + {| 𝑧 |}^{\frac{1}{3}}}) = \frac{𝑧^{- \frac{5}{6}} {\overline{𝑧}}^{- \frac{5}{6}} (1 + {| 𝑧 |}^{\frac{1}{3}}) - 𝑧^{- \frac{5}{6}} {\overline{𝑧}}^{\frac{1}{6}} 𝑧^{\frac{1}{6}} {\overline{𝑧}}^{- \frac{5}{6}}}{18 {(1 + {| 𝑧 |}^{\frac{1}{3}})}^{2}} \\ = \frac{1}{18 {| 𝑧 |}^{\frac{5}{3}} {(1 + {| 𝑧 |}^{\frac{1}{3}})}^{2}}, \end{aligned}

and a similar calculation yields

\nabla^{2} log (\frac{\sqrt{1 + {| 𝑧 - 1 |}^{\frac{1}{3}}}}{{| 𝑧 - 1 |}^{\frac{5}{6}}}) = \frac{1}{18 {| 𝑧 - 1 |}^{\frac{5}{3}} {(1 + {| 𝑧 - 1 |}^{\frac{1}{3}})}^{2}} .

Hence,

𝐾_{𝜌} (𝑧) = - \frac{1}{18} [\frac{{| 𝑧 - 1 |}^{\frac{5}{3}}}{{(1 + {| 𝑧 |}^{\frac{1}{3}})}^{3} (1 + {| 𝑧 - 1 |}^{\frac{1}{3}})} + \frac{{| 𝑧 |}^{\frac{5}{3}}}{{(1 + {| 𝑧 - 1 |}^{\frac{1}{3}})}^{3} (1 + {| 𝑧 |}^{\frac{1}{3}})}],

and that

$𝐾_{𝜌} \in 𝐶 (ℂ ∖ {0, 1})$ .
$\forall 𝑧 \in ℂ ∖ {0, 1}$ , $𝐾_{𝜌} (𝑧) < 0$ .
${lim}_{𝑧 \to 0} 𝐾_{𝜌} (𝑧) = - \frac{1}{36}$ .
${lim}_{𝑧 \to 1} 𝐾_{𝜌} (𝑧) = - \frac{1}{36}$ .
${lim}_{𝑧 \to \infty} 𝐾_{𝜌} (𝑧) = - \infty$ in any direction (as in the one-point compactification).

Hence, $\exists 𝛿 > 0$ such that $| 𝐾_{𝜌} (𝑧) + \frac{1}{36} | < \frac{1}{72}$ for any $𝑧 \in 𝐷^{*} (0, 𝛿) \cup 𝐷^{*} (1, 𝛿)$ and $\exists 𝑅 > 0$ such that $𝐾_{𝜌} (𝑧) < - 1$ for any $𝑧$ satisfying $| 𝑧 | > 𝑅$ . By compactness of $\overline{𝐷 (0, 𝑅)} ∖ (𝐷 (0, 𝛿) \cup 𝐷 (1, 𝛿))$ and continuity, it attains its supremum of some value $- 𝑀 < 0$ by Theorem 1.2.23 $Theorem 1.2.23 Theorem 1.2.23$ . Let $- 𝛽 = max {- \frac{1}{72}, - 𝑀} < 0$ .

∴ 𝐾_{𝜌} (𝑧) \leq - 𝛽 < 0 \forall 𝑧 \in ℂ ∖ {0, 1} . □

And we have the final implication:

Theorem 8.3.6 (Picard’s Little Theorem).

Let

𝑓 : ℂ \to 𝑈

be entire such that

ℂ ∖ 𝑈

contains two or more points. Then

𝑓

is constant.

Proof.

By the result of Proposition 8.3.5

Proposition 8.3.5 Proposition 8.3.5

, we may find a conformal metric

𝜌

𝑈

such that

\exists 𝛽 > 0

satisfying

𝐾_{𝜌} (𝑈) \subseteq ℝ_{\leq - 𝛽}

. Then by the aforementioned generalization of Liouville (Corollary 8.3.3.2

Corollary 8.3.3.2 Corollary 8.3.3.2

𝑓

exhibits constancy on

ℂ

and the assertion follows. ⁠

□

Remark.

This is commonly stated in its contrapositive: the image of any non-constant entire function omits at most one value.