§8.4 A Spherical Generalization of Normal Families

8.4 A Spherical Generalization of Normal Families

Picard’s Great Theorem requires a more profound concept by generalizing normal families in the one-point compactification of $ℂ$ .

Definition 8.4.1.

Let

{𝑓_{𝑛} (𝑧)}

be a (not necessarily analytic) complex function sequence on a connected set

Ω \subseteq ℂ

. If

\forall 𝐾 \subset Ω

compact,

\forall 𝑅 > 0

\exists 𝑁 \in ℕ

such that

\forall 𝑛 > 𝑁

\forall 𝑧 \in 𝐾

| 𝑓_{𝑛} (𝑧) | > 𝑅

, then $𝑓_{𝑛} ⇉ \infty$ locally uniformly spherically on $Ω$ .

When the “locally uniform limit” is taken to be $\infty$ , the condition of $𝜀$ -closeness is instead replaced by the requirement that the values eventually leave every fixed compact subset of $ℂ$ (the given definition is equivalent to: $\forall 𝐾 \subset Ω$ compact, $\forall 𝐿 \subset ℂ$ compact, $\exists 𝑁 \in ℕ$ such that $\forall 𝑛 > 𝑁$ , $\forall 𝑧 \in 𝐾$ , $𝑓_{𝑛} (𝑧) \notin 𝐿$ ). In this way, convergence to infinity is treated symmetrically with convergence to finite values by working in the Riemann sphere $\hat{ℂ}$ , where $\infty$ is simply another accumulation point.

By equipping the extended complex plane $\hat{ℂ}$ with the spherical metric instead of the Euclidean metric, convergence to $\infty$ can be treated like convergence to any finite point. In this setting, $\infty$ is simply another accumulation point, so there is no need to handle it differently from other values.

Let ${𝑎_{𝑛}}_{𝑛 \in ℕ} \subseteq \hat{ℂ}$ be a sequence. Then we say $𝑎_{𝑛} \to 𝑎_{\infty}$ spherically iff $\forall 𝜀 > 0$ , $\exists 𝑁 \in ℕ$ such that $\forall 𝑛 > 𝑁$ , $𝑑_{𝜎} (𝑎_{𝑛}, 𝑎_{\infty}) < 𝜀$ , where $𝑑_{𝜎}$ is the spherical distance.

Definition 8.4.2.

A family of meromorphic functions

ℱ︀

on some

Ω \subseteq ℂ

is said to be spherically normal iff every sequence has a locally uniformly spherically convergent subsequence on

Ω

Montel’s Theorem for holomorphically normal families in Theorem 5.2.6 $Theorem 5.2.6 Theorem 5.2.6$ can be generalized via the spherical metric by the statement of Marty’s Criterion (Theorem 8.4.6 $Theorem 8.4.6 Theorem 8.4.6$ ).

Definition 8.4.3 (Spherical Derivative).

Let $Ω \subseteq ℂ$ be an open region or domain. Suppose $𝑓 : Ω \to \hat{ℂ}$ is meromorphic. Then the spherical derivative of $𝑓$ is given by

𝑓^{♯} (𝑧) = 𝑓^{*} 𝜎 (𝑧) = \frac{2 | 𝑓^{'} (𝑧) |}{1 + {| 𝑓 (𝑧) |}^{2}}

for $𝑓 (𝑧) \neq \infty$ and

𝑓^{♯} (𝑧) = lim_{𝜁 \to 𝑧} 𝑓^{♯} (𝜁)

otherwise.

Remark.

In certain normalizations (where the Riemann sphere has north pole

(0, 0, 1)

and diameter 1), the spherical derivative will instead be defined by

𝑓^{♯} (𝑧) = \frac{| 𝑓^{'} (𝑧) |}{1 + {| 𝑓 (𝑧) |}^{2}}

Proposition 8.4.4.

Any linear fractional transformation is spherically uniformly continuous on

ℂ

Proof.

Let $𝜓 (𝑧) = \frac{𝑎 𝑧 + 𝑏}{𝑐 𝑧 + 𝑑}$ , where $𝑎 𝑑 - 𝑏 𝑐 \neq 0$ . Then,

𝜓^{'} (𝑧) = \frac{𝑎 𝑑 - 𝑏 𝑐}{{(𝑐 𝑧 + 𝑑)}^{2}} .

The spherical distance between two points $𝑤_{1} = 𝜓 (𝑧_{1}), 𝑤_{2} = 𝜓 (𝑧_{2})$ is given by

𝑑_{𝜎} (𝑤_{1}, 𝑤_{2}) = inf_{𝛾} \int_{𝛾} 𝜓^{♯} (𝑧) | d 𝑧 | = inf_{𝛾} \int_{𝛾} \frac{2 | \frac{𝑎 𝑑 - 𝑏 𝑐}{{(𝑐 𝑧 + 𝑑)}^{2}} |}{1 + {| \frac{𝑎 𝑧 + 𝑏}{𝑐 𝑧 + 𝑑} |}^{2}} | d 𝑧 |

where $𝛾$ joins $𝑧_{1}$ and $𝑧_{2}$ . The spherical distance is bounded by the integral over the Euclidean straight line $𝛾^{'}$ joining $𝑧_{1}$ and $𝑧_{2}$ :

𝑑_{𝜎} (𝑤_{1}, 𝑤_{2}) \leq \int_{𝛾^{'}} \frac{2 | 𝑎 𝑑 - 𝑏 𝑐 |}{{| 𝑐 𝑧 + 𝑑 |}^{2} + {| 𝑎 𝑧 + 𝑏 |}^{2}} | d 𝑧 | .

Since $\frac{2 | 𝑎 𝑑 - 𝑏 𝑐 |}{{| 𝑐 𝑧 + 𝑑 |}^{2} + {| 𝑎 𝑧 + 𝑏 |}^{2}} \to 0$ as $𝑧 \to \infty$ and $𝑧 \mapsto \frac{2 | 𝑎 𝑑 - 𝑏 𝑐 |}{{| 𝑐 𝑧 + 𝑑 |}^{2} + {| 𝑎 𝑧 + 𝑏 |}^{2}} \in 𝐶 (ℂ)$ , it is bounded by some constant $𝑀$ on $ℂ$ . Hence, we have

𝑑_{𝜎} (𝑤_{1}, 𝑤_{2}) \leq 𝑀 | 𝑧_{1} - 𝑧_{2} | .

Hence, $\forall 𝜀 > 0$ , $\forall | 𝑧_{1} - 𝑧_{2} | < \frac{𝜀}{𝑀}$ ,

𝑑_{𝜎} (𝜓 (𝑧_{1}), 𝜓 (𝑧_{2})) < 𝜀 . □

Proposition 8.4.5.

Let

{𝑓_{𝑛}}_{𝑛 \in ℕ}

be a sequence of holomorphic functions on a domain

Ω \subseteq ℂ

. If

𝑓_{𝑛} ⇉ 𝑓

locally uniformly spherically, then

𝑓

is either holomorphic on

Ω

or identically

\infty

Proof.

A result analogous to Theorem 2.3.6 $Theorem 2.3.6 Theorem 2.3.6$ can be used to show that $𝑓$ is spherically continuous. Let $𝑧 \in Ω$ be arbitrary.

If $𝑓 (𝑧) \neq \infty$ , then by spherical continuity, $\exists 𝛿 > 0$ such that $\forall 𝜁 \in 𝐷 (𝑧, 𝛿)$ ,
$𝑑_{𝜎} (𝑓 (𝜁), 𝑓 (𝑧)) < \frac{1}{2} 𝑑_{𝜎} (\infty, 𝑓 (𝑧)) .$
Similarly, $\exists 𝑁 \in ℕ$ such that $\forall 𝑛 > 𝑁$ ,
$𝑑_{𝜎} (𝑓 (𝜁), 𝑓_{𝑛} (𝜁)) < \frac{1}{2} 𝑑_{𝜎} (\infty, 𝑓 (𝑧)) .$
Hence, we have
$𝑑_{𝜎} (\infty, 𝑓 (𝑧)) - 𝑑_{𝜎} (𝑓 (𝑧), 𝑓_{𝑛} (𝜁)) > 0 .$
By the reverse triangle inequality, we have
$𝑑_{𝜎} (\infty, 𝑓_{𝑛} (𝜁)) > 0 .$
By Weierstrass (Theorem 4.1.1 $Theorem 4.1.1 Theorem 4.1.1$ ), $𝑓$ is holomorphic on $𝐷 (𝑧, 𝛿)$ .
Consider $𝑓 (𝑧) = \infty$ . Assume, for the sake of contradiction, $𝑧$ is an isolated pole of $𝑓$ . Hence, $\exists 𝛿$ such that $𝑓$ is holomorphic on $𝐷^{*} (𝑧, 𝛿)$ .
Because each $𝑓_{𝑛}$ is holomorphic on $𝐷 (𝑧, 𝛿)$ , by the Maximum Modulus Principle (Theorem 3.4.2 $Theorem 3.4.2 Theorem 3.4.2$ ), $\forall 𝑛 \in ℕ$ ,
$| 𝑓_{𝑛} (𝜁) | \leq sup_{𝜉 \in 𝜕 𝐷 (𝑧, 𝛿)} | 𝑓_{𝑛} (𝜉) | \forall 𝜁 \in 𝐷 (𝑧, 𝛿) .$
By letting $𝑛 \to \infty$ , we have
$| 𝑓 (𝜁) | \leq sup_{𝜉 \in 𝜕 𝐷 (𝑧, 𝛿)} | 𝑓 (𝜉) | < \infty \forall 𝜁 \in 𝐷 (𝑧, 𝛿),$
contradicting the assumption that $𝑓 (𝑧) = \infty$ is an isolated pole. Hence, $𝑧$ must be an accumulation of values evaluating to $\infty$ . By spherical continuity, $\exists 𝛿 > 0$ such that
$𝑑_{𝜎} (𝑓 (𝜁), \infty) < \frac{π}{2} \forall 𝜁 \in 𝐷 (𝑧, 𝛿) .$
Similarly, $\exists 𝑁 \in ℕ$ such that $\forall 𝑛 > 𝑁$ ,
$𝑑_{𝜎} (𝑓 (𝜁), 𝑓_{𝑛} (𝜁)) < \frac{π}{2} .$
Hence, we have
$π - 𝑑_{𝜎} (\infty, 𝑓_{𝑛} (𝜁)) = 𝑑_{𝜎} (\infty, 0) - 𝑑_{𝜎} (\infty, 𝑓_{𝑛} (𝜁)) > 0 .$
By the reverse triangle inequality, we have
$𝑑_{𝜎} (0, 𝑓_{𝑛} (𝜁)) > 0 .$
Hence each $\frac{1}{𝑓_{𝑛}}$ is holomorphic on $𝐷 (𝑧, 𝛿)$ and converges locally uniformly spherically to $\frac{1}{𝑓}$ on $𝐷 (𝑧, 𝛿)$ . By Weierstrass (Theorem 4.1.1 $Theorem 4.1.1 Theorem 4.1.1$ ), $\frac{1}{𝑓}$ is holomorphic on $𝐷 (𝑧, 𝛿)$ and has zeros that accumulate at $𝑧$ . By the Identity Theorem, $\frac{1}{𝑓} \equiv 0 ⟹ 𝑓 \equiv \infty$ on $𝐷 (𝑧, 𝛿)$ .

Let $𝑆$ be the set of all $𝑧 \in Ω$ such that $𝑓 (𝑧)$ is finite. By the argument above, $𝑆$ is open. The complement $Ω ∖ 𝑆$ then consists of all points where $𝑓 (𝑧) = \infty$ . By the argument above, $Ω ∖ 𝑆$ is also open. Since $Ω$ is connected, by Theorem 3.2.1.19 $Theorem 3.2.1.19 Theorem 3.2.1.19$ , either $𝑆 = ⌀$ or $𝑆 = Ω$ . In the former case, $𝑓 \equiv \infty$ on $Ω$ , and in the latter case, $𝑓$ is holomorphic on $Ω$ . ⁠ $□$

Theorem 8.4.6 (Marty’s Criterion).

A family of meromorphic functions $ℱ︀$ on some $Ω \subseteq ℂ$ is spherically normal iff

{𝑓^{♯} : 𝑓 \in ℱ︀},

or the family of spherical derivatives, is locally uniformly bounded in

Ω

Proof.

The condition is equivalent to that of

\frac{2 | 𝑓^{'} (𝑧) |}{1 + {| 𝑓 (𝑧) |}^{2}} \leq 𝑀 \forall 𝑓 \in ℱ︀

for all compact $𝐾 \subset Ω$ , $\forall 𝑧 \in 𝐾$ , where $𝑀$ depends only on $𝐾$ . Under the assumption that this holds, then

𝑑_{𝜎} (𝑓 (𝑧_{1}), 𝑓 (𝑧_{2})) = inf_{𝛾} \int_{𝛾} d 𝑠_{𝜎} \leq 𝑀 | 𝑧_{2} - 𝑧_{1} | \forall 𝑓 \in ℱ︀

where $𝛾$ joins $𝑓 (𝑧_{1})$ and $𝑓 (𝑧_{2})$ where $𝑧_{1}, 𝑧_{2} \in 𝐾$ . Hence, $\forall 𝜀 > 0$ , $\forall 𝑧_{1}, 𝑧_{2} \in 𝐾$ such that $| 𝑧_{1} - 𝑧_{2} | < \frac{𝜀}{𝑀}$ , $𝑑_{𝜎} (𝑓 (𝑧_{1}), 𝑓 (𝑧_{2})) < 𝜀$ , and hence $ℱ︀$ is uniformly spherically equicontinuous. Since $𝑑_{𝜎} \leq π$ for any two points by geometry of $𝑆^{2}$ , $ℱ︀$ is also uniformly spherically bounded (the compactness of $𝑆^{2}$ ). Then the Arzelà–Ascoli Theorem (Theorem 5.2.4 $Theorem 5.2.4 Theorem 5.2.4$ ) under the spherical metric gives that $ℱ︀$ is a normal family.

Conversely, assume for the sake of contradiction that $ℱ︀$ is a normal family such that conclusion is not satisfied. Then, $\exists 𝐾 \subset Ω$ compact and a sequence ${𝑓_{𝑛}}_{𝑛 \in ℕ} \subseteq ℱ︀$ such that the sequence

{sup_{𝑧 \in 𝐾} 𝑓_{𝑛}^{♯} (𝑧)}_{𝑛 \in ℕ}

tends to $\infty$ (specifically, suppose that $\forall 𝑛 \in ℕ$ , ${sup}_{𝑧 \in 𝐾} 𝑓_{𝑛}^{♯} (𝑧) > 𝑛$ ). By normality, we may extract a locally uniformly spherically convergent subsequence ${𝑓_{𝑛_{𝑘}}}_{𝑘 \in ℕ} \subseteq {𝑓_{𝑛}}_{𝑛 \in ℕ}$ . By Theorem 2.3.6 $Theorem 2.3.6 Theorem 2.3.6$ under the spherical metric, the uniform spherical limit of ${𝑓_{𝑛_{𝑘}}}_{𝑘 \in ℕ}$ , $𝑓$ , is spherically continuous on $Ω$ . For every point $𝑧 \in Ω$ , there are two possibilities:

If $𝑓 (𝑧) \neq \infty$ , then by continuity, $\exists 𝛿 > 0$ such that $\forall 𝜁 \in 𝐷 (𝑧, 𝛿)$ ,
$𝑑_{𝜎} (𝑓 (𝜁), 𝑓 (𝑧)) < \frac{1}{2} 𝑑_{𝜎} (\infty, 𝑓 (𝑧)) .$
Similarly, $\exists 𝑁 \in ℕ$ such that $\forall 𝑘 > 𝑁$ ,
$𝑑_{𝜎} (𝑓 (𝜁), 𝑓_{𝑛_{𝑘}} (𝜁)) < \frac{1}{2} 𝑑_{𝜎} (\infty, 𝑓 (𝑧)) .$
Hence, we have
$𝑑_{𝜎} (\infty, 𝑓 (𝑧)) - 𝑑_{𝜎} (𝑓 (𝑧), 𝑓_{𝑛_{𝑘}} (𝜁)) > 0 .$
By the reverse triangle inequality, we have
$𝑑_{𝜎} (\infty, 𝑓_{𝑛_{𝑘}} (𝜁)) > 0 .$
Hence, the meromorphy of each $𝑓_{𝑛_{𝑘}}$ is actually holomorphy. By continuity, $𝑓$ is locally uniformly bounded on $𝐷 (𝑧, 𝛿)$ . Hence, ${𝑓_{𝑛_{𝑘}}}_{𝑘 > 𝑁}$ locally uniformly converges on $𝐷 (𝑧, 𝛿)$ . By a result of Weierstrass (Theorem 4.1.1 $Theorem 4.1.1 Theorem 4.1.1$ ), $𝑓$ is holomorphic on $𝐷 (𝑧, 𝛿)$ and the sequence ${𝑓_{𝑛_{𝑘}}^{'}}_{𝑘 > 𝑁}$ locally uniformly converges to $𝑓^{'}$ on $𝐷 (𝑧, 𝛿)$ .
By holomorphy of $𝑓^{'}$ on $\overline{𝐷 (𝑧, \frac{𝛿}{2})}$ , $\exists 𝑀^{'} > 0$ such that ${sup}_{𝜁 \in \overline{𝐷 (𝑧, \frac{𝛿}{2})}} | 𝑓^{'} (𝜁) | < 𝑀^{'}$ . Uniform convergence of ${𝑓_{𝑛_{𝑘}}^{'}}_{𝑘 > 𝑁}$ gives the existence of some $𝑁^{'} > 𝑁$ such that $\forall 𝑘 > 𝑁^{'}$ ,
$| 𝑓_{𝑛_{𝑘}}^{'} (𝜁) - 𝑓^{'} (𝜁) | < 1 ⟹ | 𝑓_{𝑛_{𝑘}}^{'} (𝜁) | \leq 𝑀^{'} + 1 \forall 𝜁 \in \overline{𝐷 (𝑧, \frac{𝛿}{2})} .$
Therefore, ${𝑓_{𝑛_{𝑘}}^{'}}_{𝑘 > 𝑁}$ is uniformly bounded by
$𝑀 = max ({𝑀^{'} + 1} \cup {sup_{𝜁 \in \overline{𝐷 (𝑧, \frac{𝛿}{2})}} | 𝑓_{𝑛_{𝑘}}^{'} (𝜁) |}_{𝑁 < 𝑘 \leq 𝑁^{'}})$
on this compact disk. Hence, $\forall 𝑘 > 𝑁$ ,
$𝑓_{𝑛_{𝑘}}^{♯} (𝜁) = \frac{2 | 𝑓_{𝑛_{𝑘}}^{'} (𝜁) |}{1 + {| 𝑓_{𝑛_{𝑘}} (𝜁) |}^{2}} \leq 2 | 𝑓_{𝑛_{𝑘}}^{'} (𝜁) | \leq 2 𝑀 \forall 𝜁 \in 𝐷 (𝑧, \frac{𝛿}{2}) \subset \overline{𝐷 (𝑧, \frac{𝛿}{2})} .$
$𝑓 (𝑧) = \infty$ , then by continuity, $\exists 𝛿 > 0$ such that $\forall 𝜁 \in 𝐷 (𝑧, 𝛿)$ ,
$𝑑_{𝜎} (𝑓 (𝜁), \infty) < \frac{π}{2} .$
Similarly, $\exists 𝑁 \in ℕ$ such that $\forall 𝑘 > 𝑁$ ,
$𝑑_{𝜎} (𝑓 (𝜁), 𝑓_{𝑛_{𝑘}} (𝜁)) < \frac{π}{2} .$
Hence, we have
$π - 𝑑_{𝜎} (\infty, 𝑓_{𝑛_{𝑘}} (𝜁)) = 𝑑_{𝜎} (\infty, 0) - 𝑑_{𝜎} (\infty, 𝑓_{𝑛_{𝑘}} (𝜁)) > 0 .$
By the reverse triangle inequality, we have
$𝑑_{𝜎} (0, 𝑓_{𝑛_{𝑘}} (𝜁)) > 0 .$
Hence, each $𝑔_{𝑛_{𝑘}} = \frac{1}{𝑓_{𝑛_{𝑘}}}$ is holomorphic on $𝐷 (𝑧, 𝛿)$ . By continuity, $𝑔 = \frac{1}{𝑓}$ is locally uniformly bounded on $𝐷 (𝑧, 𝛿)$ . It can also be realized that ${𝑔_{𝑛_{𝑘}}}_{𝑘 > 𝑁}$ locally uniformly converges on $𝐷 (𝑧, 𝛿)$ . By a result of Weierstrass (Theorem 4.1.1 $Theorem 4.1.1 Theorem 4.1.1$ ), $𝑔$ is holomorphic on $𝐷 (𝑧, 𝛿)$ and the sequence ${𝑔_{𝑛_{𝑘}}^{'}}_{𝑘 > 𝑁}$ locally uniformly converges to $𝑔^{'}$ on $𝐷 (𝑧, 𝛿)$ .
By holomorphy of $𝑔^{'}$ on $\overline{𝐷 (𝑧, \frac{𝛿}{2})}$ , $\exists 𝑀^{'} > 0$ such that ${sup}_{𝜁 \in \overline{𝐷 (𝑧, \frac{𝛿}{2})}} | 𝑔^{'} (𝜁) | < 𝑀^{'}$ . Uniform convergence of ${𝑔_{𝑛_{𝑘}}^{'}}_{𝑘 > 𝑁}$ gives the existence of some $𝑁^{'} > 𝑁$ such that $\forall 𝑘 > 𝑁^{'}$ ,
$| 𝑔_{𝑛_{𝑘}}^{'} (𝜁) - 𝑔^{'} (𝜁) | < 1 ⟹ | 𝑔_{𝑛_{𝑘}}^{'} (𝜁) | \leq 𝑀^{'} + 1 \forall 𝜁 \in \overline{𝐷 (𝑧, \frac{𝛿}{2})} .$
Therefore, ${𝑔_{𝑛_{𝑘}}^{'}}_{𝑘 > 𝑁}$ is uniformly bounded by
$𝑀 = max ({𝑀^{'} + 1} \cup {sup_{𝜁 \in \overline{𝐷 (𝑧, \frac{𝛿}{2})}} | 𝑔_{𝑛_{𝑘}}^{'} (𝜁) |}_{𝑁 < 𝑘 \leq 𝑁^{'}})$
on this compact disk. Hence, $\forall 𝑘 > 𝑁$ ,
$𝑓_{𝑛_{𝑘}}^{♯} (𝜁) = \frac{2 | - \frac{𝑔_{𝑛_{𝑘}}^{'} (𝜁)}{𝑔_{𝑛_{𝑘}} {(𝜁)}^{2}} |}{1 + {| 𝑔_{𝑛_{𝑘}} (𝜁) |}^{- 2}} = \frac{2 | 𝑔_{𝑛_{𝑘}}^{'} (𝜁) |}{{| 𝑔_{𝑛_{𝑘}} (𝜁) |}^{2} + 1} \leq 2 | 𝑔_{𝑛_{𝑘}}^{'} (𝜁) | \leq 2 𝑀, \forall 𝜁 \in 𝐷 (𝑧, \frac{𝛿}{2}) .$

In essence, for any point $𝑧$ , there exists an open disk $𝐷_{𝑧}$ centered at $𝑧$ on which the spherical derivatives $𝑓_{𝑛_{𝑘}}^{♯}$ are bounded by some constant $𝑀_{𝑧}$ for $𝑘 > 𝑁_{𝑧}$ . By Heine–Borel (Theorem 1.1.9 $Theorem 1.1.9 Theorem 1.1.9$ ), there exists a finite collection of disks ${𝐷_{𝑧_{𝑗}}}_{1 \leq 𝑗 \leq 𝑛}$ that cover $𝐾$ . Thus, ${𝑓_{𝑛_{𝑘}}^{♯} (𝑧)}_{𝑘 > 𝑁}$ is uniformly bounded on $𝐾$ by ${max}_{1 \leq 𝑗 \leq 𝑛} 𝑀_{𝑧_{𝑗}}$ , where $𝑁 = {max}_{1 \leq 𝑗 \leq 𝑛} 𝑁_{𝑧_{𝑗}}$ , contradicting the assumption that ${sup}_{𝑧 \in 𝐾} 𝑓_{𝑛}^{♯} (𝑧) > 𝑛$ for all $𝑛 \in ℕ$ . ⁠ $□$

Theorem 8.4.7 (Fundamental Normality Test).

Let

Ω \subseteq ℂ

be a region and suppose that

ℱ︀

is a family of holomorphic functions on

Ω

. If there exist two different points

𝛼, 𝛽 \in ℂ

such that

{𝛼, 𝛽} \cap ⋃_{𝑓 \in ℱ︀} 𝑓 (Ω) = ⌀

, then

ℱ︀

must be a spherically normal family.

Proof.

Map $𝛼$ and $𝛽$ to $0, 1$ by a linear function $𝜑 (𝑧) = \frac{𝑧 - 𝛼}{𝛽 - 𝛼}$ . Then the family of holomorphic functions

\tilde{ℱ︀} = {𝜑 \circ 𝑓 : 𝑓 \in ℱ︀}

omits $0$ and $1$ for all $𝑧 \in Ω$ .

By Proposition 8.3.5 $Proposition 8.3.5 Proposition 8.3.5$ , $\exists 𝛽 > 0$ such that for

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}

as in (8.3.5) $(8.3.5) (8.3.5)$ ,

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

Therefore, if we let $𝜇 = 𝜌 \sqrt{𝛽}$ , then

𝐾_{𝜇} = - \frac{\nabla^{2} (log \circ 𝜇)}{𝜇^{2}} = - \frac{\nabla^{2} (log \circ 𝜌)}{𝜌^{2} 𝛽} = \frac{𝐾_{𝜌}}{𝛽} \leq - 1 on ℂ ∖ {0, 1} . (8.4 . 1)

Let $𝜁 \in Ω$ be arbitrary and let $𝑟 = 𝑟_{𝜁} > 0$ satisfy $𝐷 (𝜁, 𝑟_{𝜁}) \subseteq Ω$ . By Corollary 8.3.3.1 $Corollary 8.3.3.1 Corollary 8.3.3.1$ , the pullback of $𝜇$ from $ℂ ∖ {0, 1}$ to $𝐷 (𝜁, 𝑟_{𝜁}) \subseteq Ω$ satisfies

𝑓^{*} 𝜇 (𝑧) \leq 𝜆_{𝑟_{𝜁}} (𝑧 - 𝜁) ⟹ 𝜇 (𝑓 (𝑧)) | 𝑓^{'} (𝑧) | \leq \frac{2 𝑟_{𝜁}}{𝑟_{𝜁}^{2} - {| 𝑧 - 𝜁 |}^{2}}

$\forall 𝑧 \in 𝐷 (𝜁, 𝑟_{𝜁}), 𝑓 \in \tilde{ℱ︀}$ . Since $\forall 𝑤 \in ℂ ∖ {0, 1}$ ,

\frac{𝜎}{𝜇} = \frac{\frac{2}{1 + {| 𝑤 |}^{2}}}{\frac{\sqrt{1 + {| 𝑤 |}^{\frac{1}{3}}} \sqrt{1 + {| 𝑤 - 1 |}^{\frac{1}{3}}}}{{| 𝑤 |}^{\frac{5}{6}} {| 𝑤 - 1 |}^{\frac{5}{6}}}} \to {\begin{cases} 0 & as 𝑤 \to 0 \\ 0 & as 𝑤 \to 1 \\ \frac{2 {| 𝑤 |}^{- 2}}{{| 𝑤 |}^{- \frac{4}{3}}} \to 0 & as 𝑤 \to \infty \end{cases} .

Hence, there exist open neighborhoods $𝑈_{0}, 𝑈_{1}, 𝑈_{\infty}$ of $0, 1, \infty$ respectively on which $\frac{𝜎}{𝜇} < 1$ . Since $\frac{𝜎}{𝜇} \in 𝐶 (ℂ)$ , by Theorem 1.2.22 $Theorem 1.2.22 Theorem 1.2.22$ , $\exists 𝑀^{'} > 0$ such that $\frac{𝜎}{𝜇} < 𝑀^{'}$ on $ℂ ∖ (𝑈_{0} \cup 𝑈_{1} \cup 𝑈_{\infty})$ . Let $𝑀 = max (𝑀^{'}, 1)$ , and

∴ 𝜎 \leq 𝑀 𝜇 on ℂ ∖ {0, 1} .

Hence, $\forall 𝑓 \in \tilde{ℱ︀}$ , we have by virtue of (8.4.1) $(8.4.1) (8.4.1)$ ,

𝑓^{♯} (𝑧) = 𝜎 \circ 𝑓 (𝑧) | 𝑓^{'} (𝑧) | \leq 𝑀 𝜇 \circ 𝑓 (𝑧) | 𝑓^{'} (𝑧) | \leq \frac{2 𝑟_{𝜁} 𝑀}{𝑟_{𝜁}^{2} - {| 𝑧 - 𝜁 |}^{2}}

for any $𝑧 \in 𝐷 (𝜁, 𝑟_{𝜁})$ . Now restricting $𝑧$ to $𝐷 (𝜁, \frac{𝑟_{𝜁}}{2})$ , we have

{| 𝑧 - 𝜁 |}^{2} < \frac{𝑟_{𝜁}^{2}}{4} ⟹ 𝑟_{𝜁}^{2} - {| 𝑧 - 𝜁 |}^{2} > \frac{3 𝑟_{𝜁}^{2}}{4} ⟹ | 𝑓^{♯} (𝑧) | < \frac{8 𝑟_{𝜁} 𝑀}{3 𝑟_{𝜁}^{2}} = \frac{8 𝑀}{3 𝑟_{𝜁}} .

For any compact $𝐾 \subset Ω$ , the collection of open disks

{𝐷 (𝜁, \frac{𝑟_{𝜁}}{2})}_{𝜁 \in 𝐾}

forms an open cover of $𝐾$ . Hence, by Heine–Borel (Theorem 1.1.9 $Theorem 1.1.9 Theorem 1.1.9$ ), it admits a finite subcover

{𝐷 (𝜁_{𝑘}, \frac{𝑟_{𝜁_{𝑘}}}{2})}_{1 \leq 𝑘 \leq 𝑛}

for some $𝑛 \in ℕ$ . Then ${𝑓^{♯} : 𝑓 \in \tilde{ℱ︀}}$ is uniformly bounded on $𝐾$ by

𝑀_{𝐾} = max {\frac{8 𝑀}{3 𝑟_{𝜁_{𝑘}}} : 1 \leq 𝑘 \leq 𝑛}

and is thus locally uniformly bounded on $Ω$ . Marty’s Criterion (Theorem 8.4.6 $Theorem 8.4.6 Theorem 8.4.6$ ) gives the normality of $\tilde{ℱ︀}$ ; since $𝜑$ is linear, it follows that $ℱ︀$ is also normal on $Ω$ . ⁠ $□$

Corollary 8.4.7.1 (Montel–Carathéodory).

Let

Ω \subseteq ℂ

be a region and suppose that

ℱ︀

is a family of meromorphic functions on

Ω

. If there exist three different points

𝛼, 𝛽, 𝛾 \in \hat{ℂ}

such that

{𝛼, 𝛽, 𝛾} \cap ⋃_{𝑓 \in ℱ︀} 𝑓 (Ω) = ⌀

, then

ℱ︀

must be a spherically normal family.

Proof.

Let $𝜑 (𝑧) = \frac{(𝑧 - 𝛼) (𝛽 - 𝛾)}{(𝑧 - 𝛾) (𝛽 - 𝛼)}$ be a Möbius transformation mapping $𝛼, 𝛽, 𝛾$ to $0, 1, \infty$ , respectively. Hence, the family of meromorphic functions

\tilde{ℱ︀} = {𝜑 \circ 𝑓 : 𝑓 \in ℱ︀}

omits $0$ , $1$ , and $\infty$ (and hence each function is holomorphic). By the Fundamental Holomorphic Normality Test (Theorem 8.4.7 $Theorem 8.4.7 Theorem 8.4.7$ ), $\tilde{ℱ︀}$ is normal.

By Proposition 8.4.4 $Proposition 8.4.4 Proposition 8.4.4$ , $\forall 𝜀 > 0$ , $\exists 𝛿 > 0$ such that $\forall | 𝑤_{1} - 𝑤_{2} | < 𝛿$ in $ℂ$ ,

𝑑_{𝜎} (𝜑^{- 1} (𝑤_{1}), 𝜑^{- 1} (𝑤_{2})) < 𝜀 .

Let ${{\tilde{𝑓}}_{𝑛}}_{𝑛 \in ℕ}$ be any function sequence in $ℱ︀$ and let ${{\tilde{𝑓}}_{𝑛_{𝑘}}}_{𝑘 \in ℕ}$ be locally uniformly convergent to $\tilde{𝑓}$ on a compact set $𝐾 \subset Ω$ . Then $\exists 𝑁 \in ℕ$ such that $\forall 𝑘 > 𝑁$ ,

| {\tilde{𝑓}}_{𝑛_{𝑘}} (𝑧) - \tilde{𝑓} (𝑧) | < 𝛿 \forall 𝑧 \in 𝐾 .

Therefore, $\forall 𝑧 \in 𝐾$ , $𝑘 > 𝑁$ , we have

𝑑_{𝜎} (𝜑^{- 1} \circ {\tilde{𝑓}}_{𝑛_{𝑘}} (𝑧), 𝜑^{- 1} \circ \tilde{𝑓} (𝑧)) = 𝑑_{𝜎} (𝑓_{𝑛_{𝑘}} (𝑧), 𝑓 (𝑧)) < 𝜀 .

Hence, every sequence $𝑓_{𝑛}$ has a locally uniformly spherically convergent subsequence, and the normality of $ℱ︀$ follows. ⁠ $□$