§14.3 The Ahlfors–Shimizu Characteristic

14.3 The Ahlfors–Shimizu Characteristic

We now provide a second formulation of the first fundamental theorem, given by Ahlfors and Shimizu, found independently of each other.

Lemma 14.3.1.

Let $𝑈$ be a positively oriented bounded region by a piecewise $𝐶^{1}$ simple closed boundary. Let $𝑓$ be a (not identically $0$ ) holomorphic function on an open neighborhood of $\overline{𝑈}$ , $𝐺 \in 𝐶^{2} (𝑓 (\overline{𝑈}))$ . Then

𝐼 = \oint_{𝜕 𝑈} \nabla_{\hat{𝐧}} 𝐺 (| 𝑓 (𝑧) |) | d 𝑧 | = \int_{𝑈} 𝑔 (| 𝑓 (𝑧) |) {| 𝑓^{'} (𝑧) |}^{2} 𝑟 d 𝑟 \land d 𝜃, (14.3 . 1)

where

𝑧 = 𝑟 e^{i 𝜃}

𝑔 (𝑅) = 𝐺^{″} (𝑅) + \frac{1}{𝑅} 𝐺^{'} (𝑅)

, and

\hat{𝐧}

is the unit normal vector pointing towards the exterior of

𝜕 𝑈

Proof.

Let the zeros of $𝑓$ in $𝑈$ (which are finite, otherwise they accumulate) be ${𝑧_{𝑘}}_{𝑘 = 1}^{𝑛}$ . Choose $𝜀^{'} > 0$ so that for any $0 < 𝜀 < 𝜀^{'}$ the disks ${𝐷 (𝑧_{𝑘}, 𝜀)}_{𝑘 = 1}^{𝑛}$ each lie in the open set $𝑈$ and are pairwise disjoint from one another (we do this to ensure $𝑔$ is continuous, as $𝑔$ is singular when $𝑅 = | 𝑓 (𝑧) | = 0$ ).

Applying Green’s Theorem (Theorem 1.2.10 $Theorem 1.2.10 Theorem 1.2.10$ )¹

(\oint_{𝜕 𝑈} + \sum_{𝑘 = 1}^{𝑛} \oint_{𝜕 𝐷 (𝑧_{𝑘}, 𝜀)}) 𝐯 \cdot \hat{𝐧} d 𝑠 = \iint_{𝑈 ∖ ⋃ 𝐷 (𝑧_{𝑘}, 𝜀)} 𝛁 \cdot 𝐯 d 𝐴

to $𝐼$ (so letting $𝐯 = 𝛁 𝐺 (| 𝑓 (𝑧) |)$ ), we have

\begin{aligned} 𝐼 & = (\oint_{𝜕 𝑈} + \sum_{𝑘 = 1}^{𝑛} \oint_{𝜕 𝐷 (𝑧_{𝑘}, 𝜀)}) 𝛁 𝐺 (| 𝑓 (𝑧) |) \cdot \hat{𝐧} | d 𝑧 | \\ = \iint_{𝑈 ∖ ⋃ 𝐷 (𝑧_{𝑘}, 𝜀)} \nabla^{2} 𝐺 (| 𝑓 (𝑧) |) 𝑟 d 𝑟 d 𝜃 . (14.3 . 2) \end{aligned}

Letting $𝑅 = | 𝑓 (𝑧) | = \sqrt{𝑓 (𝑧) \overline{𝑓 (𝑧)}}$ and adopting the elliptical notations $𝜕_{𝑧} = 𝜕 / 𝜕 𝑧$ , $𝜕_{\overline{𝑧}} = 𝜕 / 𝜕 \overline{𝑧}$ etc.,

𝜕_{𝑧} 𝑅 = \frac{𝜕_{𝑧} (𝑓 \overline{𝑓})}{2 𝑅} = \frac{𝑓^{'} \overline{𝑓} + 𝑓 \overline{𝜕_{\overline{𝑧}} 𝑓}}{2 𝑅} = \frac{𝑓^{'} \overline{𝑓}}{2 𝑅}, 𝜕_{\overline{𝑧}} 𝑅 = \frac{𝜕_{\overline{𝑧}} (𝑓 \overline{𝑓})}{2 𝑅} = \frac{𝑓 \overline{𝑓^{'}}}{2 𝑅},

and

\begin{matrix} (𝜕_{𝑧} 𝑅) (𝜕_{\overline{𝑧}} 𝑅) = \frac{{| 𝑓 |}^{2} {| 𝑓^{'} |}^{2}}{4 𝑅^{2}} = \frac{{| 𝑓^{'} |}^{2}}{4}, \\ 𝜕_{𝑧 \overline{𝑧}} 𝑅 = 𝜕_{\overline{𝑧}} (\frac{𝑓^{'} \overline{𝑓}}{2 𝑅}) = \frac{2 𝑓^{'} \overline{𝑓^{'}} 𝑅 - 2 𝑓^{'} \overline{𝑓} (\frac{𝑓 \overline{𝑓^{'}}}{2 𝑅})}{4 𝑅^{2}} = \frac{2 {| 𝑓^{'} |}^{2} 𝑅^{2} - {| 𝑓^{'} 𝑓 |}^{2}}{4 𝑅^{3}} = \frac{{| 𝑓^{'} |}^{2}}{4 𝑅}, \end{matrix}

since

𝜕_{\overline{𝑧}} 𝐺 (𝑅) = 𝐺^{'} (𝑅) 𝜕_{\overline{𝑧}} 𝑅 and 𝜕_{𝑧} 𝐺^{'} (𝑅) = 𝐺^{″} (𝑅) 𝜕_{𝑧} 𝑅,

\begin{aligned} 𝜕_{𝑧 \overline{𝑧}} 𝐺 (𝑅) = 𝜕_{𝑧} 𝐺^{'} (𝑅) 𝜕_{\overline{𝑧}} 𝑅 + 𝐺^{'} (𝑅) 𝜕_{\overline{𝑧} 𝑧} 𝑅 & = 𝐺^{″} (𝑅) \frac{𝑓^{'} \overline{𝑓}}{2 𝑅} \cdot \frac{𝑓 \overline{𝑓^{'}}}{2 𝑅} + 𝐺^{'} (𝑅) \frac{{| 𝑓^{'} |}^{2}}{4 𝑅} \\ = \frac{{| 𝑓^{'} |}^{2}}{4} (𝐺^{″} (𝑅) + \frac{𝐺^{'} (𝑅)}{𝑅}), \end{aligned}

it follows that $\nabla^{2} 𝐺 (| 𝑓 (𝑧) |) = 𝑔 (𝑅) {| 𝑓^{'} |}^{2}$ . Substituting this in (14.3.2) $(14.3.2) (14.3.2)$ gives

(\oint_{𝜕 𝑈} + \sum_{𝑘 = 1}^{𝑛} \oint_{𝜕 𝐷 (𝑧_{𝑘}, 𝜀)}) 𝛁 𝐺 (| 𝑓 (𝑧) |) \cdot \hat{𝐧} | d 𝑧 | = \iint_{𝑈 ∖ ⋃ 𝐷 (𝑧_{𝑘}, 𝜀)} \nabla^{2} 𝐺 (| 𝑓 (𝑧) |) 𝑟 d 𝑟 d 𝜃 .

As $𝜀 \to 0^{+}$ , the right-hand side is simply the desired quantity in (14.3.1) $(14.3.1) (14.3.1)$ . By the continuous differentiability of $𝐺$ , $\exists 𝑀 > 0$ such that $| 𝛁 𝐺 \cdot \hat{𝐧} | \leq 𝑀$ on $⋃_{𝑘} \overline{𝐷 (𝑧_{𝑘}, 𝜀)}$ , thus

| \oint_{𝜕 𝐷 (𝑧_{𝑘}, 𝜀)} 𝛁 𝐺 \cdot \hat{𝐧} d 𝑠 | \leq \int_{0}^{2 π} 𝑀 𝜀 d 𝜃 \to 0 as 𝜀 \to 0^{+} .

Thus, the left-hand side expression of (14.3.2) $(14.3.2) (14.3.2)$ also tends to that of (14.3.1) $(14.3.1) (14.3.1)$ . ⁠ $□$

We apply Lemma 14.3.1 $Lemma 14.3.1 Lemma 14.3.1$ to

𝐺 (𝑅) = log (\frac{\sqrt{1 + 𝑅^{2}}}{2}) ⟹ 𝑔 (𝑅) = \frac{1 - 𝑅^{2}}{{(1 + 𝑅^{2})}^{2}} + \frac{1}{𝑅} (\frac{𝑅}{1 + 𝑅^{2}}) = \frac{2}{{(1 + 𝑅^{2})}^{2}} .

Let $𝑓$ be meromorphic on $\overline{𝐷 (0, 𝑟)}$ with no poles on the boundary, and let its poles in the interior be at each ${𝑏_{𝑗}}_{𝑗 = 1}^{𝑛}$ of respective orders ${𝑘_{𝑗}}_{𝑗 = 1}^{𝑛}$ . Let $𝜀^{'} > 0$ such that $\forall 0 < 𝜀 < 𝜀^{'}$ , the disks $\overline{𝐷 (𝑏_{𝑗}, 𝜀)}$ are disjoint and all lie in $𝐷 (0, 𝑟)$ . We aim to apply the lemma on this multiply connected region on which $𝑓$ is holomorphic.

In a prescribed small disk containing a pole (fix $𝑗$ ), $𝑓$ has the Laurent series

𝑓 (𝑧) = \sum_{𝑖 = - 𝑘_{𝑗}}^{\infty} 𝑐_{𝑖} {(𝑧 - 𝑏_{𝑗})}^{𝑖} = \frac{𝜙 (𝑧)}{{(𝑧 - 𝑏_{𝑗})}^{𝑘_{𝑗}}},

where $𝜙$ is analytic and non-vanishing in the closed disk (choose $𝜀$ such that $1 / 𝑓$ is analytic therein too). On the boundary $𝜕 𝐷 (𝑏_{𝑗}, 𝜀)$ ,

| 𝑓 (𝑧) | = | 𝜙 (𝑧) | 𝜀^{- 𝑘_{𝑗}} .

This implies that

\begin{aligned} 𝐺 (| 𝑓 (𝑧) |) & = log | 𝑓 (𝑧) | + log \sqrt{1 + \frac{1}{{| 𝑓 (𝑧) |}^{2}}} - log 2 \\ = log | 𝜙 (𝑧) | - 𝑘_{𝑗} log 𝜀 + log \sqrt{1 + \frac{1}{{| 𝑓 (𝑧) |}^{2}}} - log 2 . \end{aligned}

On the contour defined by $𝜕 𝐷 (𝑏_{𝑗}, 𝜀)$ ²,

\nabla_{\hat{𝐧}} 𝐺 (| 𝑓 (𝑧) |) = - \frac{𝜕}{𝜕 𝜀} 𝐺 (| 𝑓 (𝑧) |) = \frac{𝑘_{𝑗}}{𝜀} - \frac{\frac{𝜕}{𝜕 𝜀} | 𝜙 (𝑧) |}{| 𝜙 (𝑧) |} - \frac{\frac{𝜕}{𝜕 𝜀} (\frac{1}{{| 𝑓 (𝑧) |}^{2}})}{2 + \frac{2}{{| 𝑓 (𝑧) |}^{2}}} = \frac{𝑘_{𝑗}}{𝜀} + 𝒪︀ (1)

as $𝜀 \to 0^{+}$ , where $𝒪︀ (1)$ is uniform in $𝑡$ (for $𝑧 = 𝑏_{𝑗} + 𝜀 e^{i 𝑡}$ ). By the lemma,

\begin{aligned} (\oint_{𝜕 𝐷 (0, 𝑟)} + \sum_{𝑗 = 1}^{𝑛} \oint_{𝜕 𝐷 (𝑏_{𝑗}, 𝜀)}) \nabla_{\hat{𝐧}} 𝐺 (| 𝑓 (𝑧) |) | d 𝑧 | \\ \begin{matrix}  \end{matrix} = \iint_{𝐷 (0, 𝑟) ∖ ⋃ 𝐷 (𝑏_{𝑗}, 𝜀)} \frac{2 {| 𝑓^{'} (𝑧) |}^{2}}{{(1 + {| 𝑓 (𝑧) |}^{2})}^{2}} d 𝐴, \end{aligned}

which requires the computation

\oint_{𝜕 𝐷 (𝑏_{𝑗}, 𝜀)} (\frac{𝑘_{𝑗}}{𝜀} + 𝒪︀ (1)) | d 𝑧 | = \int_{0}^{2 π} (𝑘_{𝑗} + 𝜀 \cdot 𝒪︀ (1)) d 𝜃 = 2 π 𝑘_{𝑗} + 𝒪︀ (𝜀) .

Taking the limit $𝜀 \to 0^{+}$ hence gives

\begin{aligned} \frac{1}{2 π} \int_{0}^{2 π} \nabla_{\hat{𝐧}} 𝐺 (| 𝑓 (𝑟 e^{i 𝜃}) |) 𝑟 d 𝜃 + 𝑛 (𝑟, 𝑓) \\ \begin{matrix}  \end{matrix} = \frac{1}{4 π} \iint_{𝐷 (0, 𝑟)} \frac{4 {| 𝑓^{'} (𝑧) |}^{2}}{{(1 + {| 𝑓 (𝑧) |}^{2})}^{2}} d 𝐴 = \frac{1}{4 π} \iint_{𝐷 (0, 𝑟)} 𝑓^{♯} {(𝑧)}^{2} d 𝐴 \end{aligned}

Let the expression on the right-hand side be denoted by $𝐴 (𝑟, 𝑓)$ . We thus derive

𝐴 (𝑟, 𝑓) = \frac{𝑟}{2 π} \int_{0}^{2 π} \frac{𝜕}{𝜕 𝑟} log (\frac{\sqrt{1 + {| 𝑓 (𝑟 e^{i 𝜃}) |}^{2}}}{2}) d 𝜃 + 𝑛 (𝑟, 𝑓) .

By dividing by $𝑟$ , changing variables, and integrating from $0$ to $𝑟$ , we have

\begin{aligned} \int_{0}^{𝑟} \frac{𝐴 (𝑡, 𝑓) d 𝑡}{𝑡} & = lim_{𝜀 \to 0^{+}} [\int_{𝜀}^{𝑟} \frac{1}{2 π} \frac{d}{d 𝑡} \int_{0}^{2 π} log (\frac{\sqrt{1 + {| 𝑓 (𝑡 e^{i 𝜃}) |}^{2}}}{2}) d 𝜃 d 𝑡 \\ \begin{matrix}  \end{matrix} + \int_{𝜀}^{𝑟} \frac{𝑛 (0, 𝑓)}{𝑡} d 𝑡] + 𝑁 (𝑟, 𝑓) - 𝑛 (0, 𝑓) log 𝑟 . \\ = lim_{𝜀 \to 0^{+}} \int_{𝜀}^{𝑟} \frac{𝑛 (0, 𝑓)}{𝑡} d 𝑡 - \frac{1}{2 π} \int_{0}^{2 π} log (\frac{\sqrt{1 + {| 𝑓 (𝜀 e^{i 𝜃}) |}^{2}}}{2}) d 𝜃 \\ \begin{matrix}  \end{matrix} + 𝑁 (𝑟, 𝑓) - 𝑛 (0, 𝑓) log 𝑟 + \frac{1}{2 π} \int_{0}^{2 π} log (\frac{\sqrt{1 + {| 𝑓 (𝑟 e^{i 𝜃}) |}^{2}}}{2}) d 𝜃 . (14.3 . 3) \end{aligned}

The limit expression of (14.3.3) $(14.3.3) (14.3.3)$ is written in its present form to ensure convergence in the event of a pole.

Definition 14.3.2 (Ahlfors–Shimizu Proximity Function).

For $𝑎 \in \hat{ℂ}$ , define the Ahlfors–Shimizu proximity function to be equal to:

𝑚_{0} (𝑟, 𝑎, 𝑓) = \frac{1}{2 π} \int_{0}^{2 π} log (\frac{1}{𝑘 (𝑓 (𝑟 e^{i 𝜃}), 𝑎)}) d 𝜃 (14.3 . 4)

for finite $𝑎$ and

𝑚_{0} (𝑟, 𝑓) = \frac{1}{2 π} \int_{0}^{2 π} log (\frac{1}{𝑘 (𝑓 (𝑟 e^{i 𝜃}), \infty)}) d 𝜃

for $𝑎 = \infty$ , where

𝑘 (𝑤, 𝑎) = {\begin{cases} \frac{2 | 𝑤 - 𝑎 |}{\sqrt{(1 + {| 𝑤 |}^{2}) (1 + {| 𝑎 |}^{2})}} & 𝑎 \neq \infty \land 𝑤 \neq \infty, \\ \frac{2}{\sqrt{1 + {| 𝑤 |}^{2}}} & 𝑎 = \infty \land 𝑤 \neq \infty, \\ \frac{2}{\sqrt{1 + {| 𝑎 |}^{2}}} & 𝑤 = \infty \land 𝑎 \neq \infty, \\ 0 & 𝑤 = \infty \land 𝑎 = \infty . \end{cases} (14.3 . 5)

Definition 14.3.3 (Ahlfors–Shimizu Characteristic Function).

The Ahlfors–Shimizu characteristic function $𝑇_{0} (𝑟, 𝑓)$ is defined by

𝑇_{0} (𝑟, 𝑓) = \int_{0}^{𝑟} \frac{𝐴 (𝑡, 𝑓) d 𝑡}{𝑡} .

Theorem 14.3.4.

For $𝑓$ meromorphic on a neighborhood of $\overline{𝐷 (0, 𝑟)}$ , then

𝑇_{0} (𝑟, 𝑓) = 𝑁 (𝑟, 𝑓) + 𝑚_{0} (𝑟, 𝑓) - 𝑚_{0} (0, 𝑓)

for $𝑓 (0) \neq \infty$ , and

𝑇_{0} (𝑟, 𝑓) = 𝑁 (𝑟, 𝑓) + 𝑚_{0} (𝑟, 𝑓) - log | \frac{𝑐}{2} |

𝑓 (0) = \infty

, where for some

𝑘 < 0

𝑐 \in ℂ^{*}

satisfies

𝑓 (𝑧) = 𝑐 𝑧^{𝑘} + 𝒪︀ (𝑧^{𝑘 + 1})

𝑧 \to 0

Proof.

In the first case for $𝑓 (0) \neq \infty$ , from (14.3.3) $(14.3.3) (14.3.3)$ , we have

\begin{aligned} \int_{0}^{𝑟} \frac{𝐴 (𝑡, 𝑓) d 𝑡}{𝑡} & = \frac{1}{2 π} \int_{0}^{2 π} log (\frac{\sqrt{1 + {| 𝑓 (𝑟 e^{i 𝜃}) |}^{2}}}{2}) d 𝜃 \\ \begin{matrix}  \end{matrix} - \frac{1}{2 π} \int_{0}^{2 π} log (\frac{\sqrt{1 + {| 𝑓 (0) |}^{2}}}{2}) d 𝜃 + 𝑁 (𝑟, 𝑓) \\ 𝑇_{0} (𝑟, 𝑓) & = 𝑁 (𝑟, 𝑓) + 𝑚_{0} (𝑟, 𝑓) - 𝑚_{0} (0, 𝑓) . (14.3 . 6) \end{aligned}

Now assume a pole; let $𝑓 (𝑧) = 𝑐 𝑧^{𝑘} + 𝒪︀ (𝑧^{𝑘 + 1})$ , where $𝑘 < 0$ and $𝑐 \neq 0$ . It follows that

\begin{matrix} | 𝑓 (𝜀 e^{i 𝜃}) | = | 𝑐 | 𝜀^{𝑘} + 𝒪︀ (𝜀^{𝑘 + 1}) = | 𝑐 | 𝜀^{𝑘} (1 + 𝒪︀ (𝜀)) \\ log | 𝑓 (𝜀 e^{i 𝜃}) | = log | 𝑐 | + 𝑘 log 𝜀 + log (1 + 𝒪︀ (𝜀)) \end{matrix}

where the errors are uniform in $𝜃$ . Since

\begin{aligned} log (\frac{\sqrt{1 + {| 𝑓 (𝜀 e^{i 𝜃}) |}^{2}}}{2}) & = log | 𝑓 (𝜀 e^{i 𝜃}) | + log \sqrt{1 + \frac{1}{{| 𝑓 (𝜀 e^{i 𝜃}) |}^{2}}} - log 2 \\ = log | 𝑓 (𝜀 e^{i 𝜃}) | + 𝒪︀ ({| 𝑓 (𝜀 e^{i 𝜃}) |}^{- 2}) - log 2 \\ = log | \frac{𝑐}{2} | + 𝑘 log 𝜀 + 𝒪︀ (𝜀), \end{aligned}

it follows from $𝑘 = - 𝑛 (0, 𝑓)$ , that

\begin{aligned} \int_{𝜀}^{𝑟} \frac{𝑛 (0, 𝑓)}{𝑡} d 𝑡 - \frac{1}{2 π} \int_{0}^{2 π} log (\frac{\sqrt{1 + {| 𝑓 (𝜀 e^{i 𝜃}) |}^{2}}}{2}) d 𝜃 \\ = - 𝑘 log 𝑟 - log | 𝑐 | + 𝒪︀ (𝜀) + log 2 \\ \to 𝑛 (0, 𝑓) log 𝑟 - log | \frac{𝑐}{2} | as 𝜀 \to 0 . \end{aligned}

Hence, from (14.3.3) $(14.3.3) (14.3.3)$ ,

\begin{aligned} \int_{0}^{𝑟} \frac{𝐴 (𝑡, 𝑓) d 𝑡}{𝑡} & = 𝑁 (𝑟, 𝑓) - log | \frac{𝑐}{2} | \\ \begin{matrix}  \end{matrix} + \frac{1}{2 π} \int_{0}^{2 π} log (\frac{\sqrt{1 + {| 𝑓 (𝑟 e^{i 𝜃}) |}^{2}}}{2}) d 𝜃 \\ 𝑇_{0} (𝑟, 𝑓) & = 𝑁 (𝑟, 𝑓) + 𝑚_{0} (𝑟, 𝑓) - log | \frac{𝑐}{2} | . □ (14.3 . 7) \end{aligned}

Theorem 14.3.5 (First Fundamental Theorem in Ahlfors–Shimizu Form).

Let $𝑓$ be meromorphic on $𝐷 (0, 𝑅)$ (where $0 < 𝑅 \leq \infty$ ). For $0 < 𝑟 < 𝑅$ and $𝑎 \in \hat{ℂ}$ such that $𝑓 (0) \neq 𝑎$ ,

𝑇_{0} (𝑟, 𝑓) = 𝑁 (𝑟, 𝑎, 𝑓) + 𝑚_{0} (𝑟, 𝑎, 𝑓) - 𝑚_{0} (0, 𝑎, 𝑓) .

If $𝑓 (0) = 𝑎 \neq \infty$ , then

𝑇_{0} (𝑟, 𝑓) = 𝑁 (𝑟, 𝑎, 𝑓) + 𝑚_{0} (𝑟, 𝑎, 𝑓) - log | \frac{1 + {| 𝑎 |}^{2}}{2 𝑐_{𝑎}} |,

where $𝑐_{𝑎}$ is the first nonzero coefficient of the Laurent expansion of $𝑓 - 𝑎$ . If $𝑓 (0) = 𝑎 = \infty$ , then

𝑇_{0} (𝑟, 𝑓) = 𝑁 (𝑟, 𝑎, 𝑓) + 𝑚_{0} (𝑟, 𝑎, 𝑓) - log | \frac{𝑐}{2} |

where

𝑐

is the first non-zero coefficient of the Laurent series of

𝑓

Proof.

For $𝑎 \neq \infty$ , consider the Möbius transformation $𝑊 = \frac{1 + \overline{𝑎} 𝑤}{𝑤 - 𝑎}$ , where $𝑤 = 𝑓 (𝑧)$ , and denote the function $𝑊 = 𝐹 (𝑧)$ . Then

\begin{aligned} | 𝐹^{'} (𝑧) | & = | \frac{d 𝑊}{d 𝑤} \frac{d 𝑤}{d 𝑧} | = | \frac{\overline{𝑎} (𝑤 - 𝑎) - (1 + \overline{𝑎} 𝑤)}{{(𝑤 - 𝑎)}^{2}} | | 𝑓^{'} (𝑧) | = \frac{1 + {| 𝑎 |}^{2}}{{| 𝑤 - 𝑎 |}^{2}} | 𝑓^{'} (𝑧) | \\ \frac{2 | 𝐹^{'} (𝑧) |}{1 + {| 𝐹 (𝑧) |}^{2}} & = \frac{2 \frac{1 + {| 𝑎 |}^{2}}{{| 𝑤 - 𝑎 |}^{2}}}{1 + {| \frac{1 + \overline{𝑎} 𝑤}{𝑤 - 𝑎} |}^{2}} | \frac{d 𝑤}{d 𝑧} | = \frac{2 {| 𝑤 - 𝑎 |}^{2}}{{| 𝑤 - 𝑎 |}^{2} + {| 1 + \overline{𝑎} 𝑤 |}^{2}} \frac{1 + {| 𝑎 |}^{2}}{{| 𝑤 - 𝑎 |}^{2}} | \frac{d 𝑤}{d 𝑧} | \\ = 2 \frac{{| 𝑎 |}^{2} + 1}{(𝑤 - 𝑎) (\overline{𝑤} - \overline{𝑎}) + (1 + \overline{𝑎} 𝑤) (1 + 𝑎 \overline{𝑤})} | \frac{d 𝑤}{d 𝑧} | \\ = 2 \frac{{| 𝑎 |}^{2} + 1}{{| 𝑤 |}^{2} + {| 𝑎 |}^{2} - 𝑎 \overline{𝑤} - \overline{𝑎} 𝑤 + 1 + {| 𝑎 |}^{2} {| 𝑤 |}^{2} + \overline{𝑎} 𝑤 + 𝑎 \overline{𝑤}} | \frac{d 𝑤}{d 𝑧} | \\ = \frac{2 ({| 𝑎 |}^{2} + 1)}{({| 𝑤 |}^{2} + 1) ({| 𝑎 |}^{2} + 1)} | \frac{d 𝑤}{d 𝑧} | = \frac{2 | 𝑓^{'} (𝑧) |}{1 + {| 𝑓 (𝑧) |}^{2}} . (†) \end{aligned}

In other words, $𝑓^{♯} \equiv 𝐹^{♯}$ (where $𝑓^{♯} = \frac{2 | 𝑓^{'} |}{1 + {| 𝑓 |}^{2}}$ is the spherical derivative as in Definition 8.4.3 $Definition 8.4.3 Definition 8.4.3$ ), implying that

𝐴 (𝑟, 𝑓) = \frac{1}{4 π} \iint_{𝐷 (0, 𝑟)} 𝑓^{♯} {(𝑧)}^{2} d 𝐴 = \frac{1}{4 π} \iint_{𝐷 (0, 𝑟)} 𝐹^{♯} {(𝑧)}^{2} d 𝐴 = 𝐴 (𝑟, 𝐹),

and consequently,

𝑇_{0} (𝑟, 𝑓) = \int_{0}^{𝑟} \frac{𝐴 (𝑡, 𝑓)}{𝑡} d 𝑡 = \int_{0}^{𝑟} \frac{𝐴 (𝑡, 𝐹)}{𝑡} d 𝑡 = 𝑇_{0} (𝑟, 𝐹) . (a)

If $𝑧$ is a solution to $𝑓 (𝑧) = 𝑎$ of order $𝑘$ , then $1 + \overline{𝑎} 𝑤 = 1 + {| 𝑎 |}^{2} \geq 1$ . Then $𝑊 = 𝐹 (𝑧)$ has a pole of order $𝑘$ . Thus,

𝑁 (𝑟, 𝐹) \equiv 𝑁 (𝑟, 𝑎, 𝑓) . (b)

From ( $†$ ), since $𝑘 (𝑤, 𝑎) = \frac{2 | 𝑤 - 𝑎 |}{\sqrt{(1 + {| 𝑤 |}^{2}) (1 + {| 𝑎 |}^{2})}}$ ,

𝑓^{♯} (𝑧) = 𝐹^{♯} (𝑧) = {𝑘 (𝑤, 𝑎)}^{2} \frac{{| 𝑎 |}^{2} + 1}{2 {| 𝑤 - 𝑎 |}^{2}} = \frac{| 𝐹^{'} (𝑧) | {𝑘 (𝑤, 𝑎)}^{2}}{2},

we arrive at

𝑘 (𝑤, 𝑎) = \frac{2}{\sqrt{1 + {| 𝑊 |}^{2}}} = 𝑘 (𝑊, \infty) .

Then,

\begin{aligned} 𝑚_{0} (𝑟, 𝑎, 𝑓) & = \int_{0}^{2 π} log (\frac{1}{𝑘 (𝑓 (𝑟 e^{i 𝜃}), 𝑎)}) d 𝜃 \\ = \int_{0}^{2 π} log (\frac{1}{𝑘 (𝐹 (𝑟 e^{i 𝜃}), \infty)}) d 𝜃 = 𝑚_{0} (𝑟, 𝐹) . (c) \end{aligned}

Use $(a)$ $Part 3 Part 3$ , $(b)$ $Part 3 Part 3$ , and $(c)$ $Part 3 Part 3$ , and apply Theorem 14.3.4 $Theorem 14.3.4 Theorem 14.3.4$ to $𝐹$ .

If $𝑓 (0) \neq 𝑎 (\neq \infty)$ , since $𝐹 (0) = \frac{1 + \overline{𝑎} 𝑓 (0)}{𝑓 (0) - 𝑎}$ is finite,
$\begin{aligned} 𝑇_{0} (𝑟, 𝐹) & = 𝑁 (𝑟, 𝐹) + 𝑚_{0} (𝑟, 𝐹) - 𝑚_{0} (0, 𝐹) \\ = 𝑇_{0} (𝑟, 𝑓) = 𝑁 (𝑟, 𝑎, 𝑓) + 𝑚_{0} (𝑟, 𝑎, 𝑓) - 𝑚_{0} (0, 𝑎, 𝑓), \end{aligned}$
If $𝑓 (0) = 𝑎 \neq \infty$ , then $𝐹 (0) = \infty$ and $𝑓 (𝑧) = 𝑎 + 𝑐_{𝑎} 𝑧^{𝑘} + 𝒪︀ (𝑧^{𝑘 + 1})$ for some $𝑘 \in ℕ$ . Then
$𝐹 (𝑧) = \frac{1 + \overline{𝑎} 𝑎 + 𝒪︀ (𝑧^{𝑘})}{𝑐_{𝑎} 𝑧^{𝑘} + 𝒪︀ (𝑧^{𝑘 + 1})}$
which implies that
$𝐹 (𝑧) 𝑧^{𝑘} = \frac{1 + {| 𝑎 |}^{2} + 𝒪︀ (𝑧^{𝑘})}{𝑐_{𝑎} + 𝒪︀ (𝑧)} \to \frac{1 + {| 𝑎 |}^{2}}{𝑐_{𝑎}} as 𝑧 \to 0,$
so the leading coefficient of the Laurent expansion of $𝐹$ at 0 is $𝐶 = \frac{1 + {| 𝑎 |}^{2}}{𝑐_{𝑎}}$ . By applying Theorem 14.3.4 $Theorem 14.3.4 Theorem 14.3.4$ to $𝐹$ , we have
$\begin{aligned} 𝑇_{0} (𝑟, 𝐹) & = 𝑁 (𝑟, 𝐹) + 𝑚_{0} (𝑟, 𝐹) - log | \frac{𝐶}{2} | \\ = 𝑇_{0} (𝑟, 𝑓) = 𝑁 (𝑟, 𝑎, 𝑓) + 𝑚_{0} (𝑟, 𝑎, 𝑓) - log | \frac{1 + {| 𝑎 |}^{2}}{2 𝑐_{𝑎}} | . \end{aligned}$
If $𝑓 (0) \neq 𝑎 = \infty$ , then the assertion follows directly from Theorem 14.3.4 $Theorem 14.3.4 Theorem 14.3.4$ on $𝑓$ .
If $𝑓 (0) = 𝑎 = \infty$ , the assertion also follows from Theorem 14.3.4 $Theorem 14.3.4 Theorem 14.3.4$ on $𝑓$ . $□$

Theorem 14.3.6.

Let

𝑓 : 𝐷 (0, 𝑅) \to \hat{ℂ}

be meromorphic. Then for each

0 < 𝑟 < 𝑅

, the Nevanlinna characteristic

𝑇

and the Ahlfors–Shimizu characteristic

𝑇_{0}

differ by a term that is uniformly bounded in

𝑟

Proof.

Observe that for $𝑧 \in 𝜕 𝐷 (0, 𝑟)$ ,

{log}^{+} | 𝑓 (𝑧) | - log 2 \leq log (\frac{\sqrt{1 + {| 𝑓 (𝑧) |}^{2}}}{2}) = \frac{1}{2} log (1 + {| 𝑓 (𝑧) |}^{2}) - log 2 .

Since $1 + 𝑥^{2} \leq max {2, 2 𝑥^{2}}$ for any real $𝑥$ ,

\begin{matrix} \frac{1}{2} log (1 + {| 𝑓 (𝑧) |}^{2}) - log 2 \leq \frac{1}{2} (max {log 2, log 2 + 2 log | 𝑓 (𝑧) |}) - log 2 \\ \leq {log}^{+} | 𝑓 (𝑧) | - \frac{1}{2} log 2 . \end{matrix}

Integrating and adding $𝑁$ , we have

𝑇 (𝑟, 𝑓) - log 2 \leq 𝑚_{0} (𝑟, 𝑓) + 𝑁 (𝑟, 𝑓) \leq 𝑇 (𝑟, 𝑓) - \frac{1}{2} log 2 .

By the First Fundamental Theorem (Theorem 14.3.5 $Theorem 14.3.5 Theorem 14.3.5$ ) in Ahlfors–Shimizu Form with $𝑎 = \infty$ , if we let

𝐶 = {\begin{cases} 𝑚_{0} (0, 𝑎, 𝑓) & if 𝑓 (0) \neq \infty \\ log | \frac{𝑐}{2} | & if 𝑓 (0) = \infty \end{cases},

where $𝑐$ is the first nonzero coefficient of the Laurent expansion of $𝑓$ , it follows that

𝑇 (𝑟, 𝑓) - 𝐶 - log 2 \leq 𝑇_{0} (𝑟, 𝑓) \leq 𝑇 (𝑟, 𝑓) - \frac{1}{2} log 2 - 𝐶,

where

𝐶

does not depend on

𝑅

. ⁠

□

Let $𝑆$ be the Riemann sphere centered at $(0, 0, 0)$ , radius 1, with a stereographic projection with center $(0, 0, 1)$ as in Section 2.1 $Section 2.1 Section 2.1$ . Letting

𝝈 (𝑤) = \frac{1}{{| 𝑤 |}^{2} + 1} (2 ℜ𝔢 𝑤, 2 ℑ𝔪 𝑤, {| 𝑤 |}^{2} - 1) = \frac{(2 𝑢, 2 𝑣, 𝑢^{2} + 𝑣^{2} - 1)}{𝑢^{2} + 𝑣^{2} + 1},

for $𝑤 = 𝑢 + i 𝑣$ , then for two points $𝑤_{1}, 𝑤_{2} \in \hat{ℂ}$ , their spherical points are $𝝈 (𝑤_{1}), 𝝈 (𝑤_{2})$ . The Euclidean straight-line distance between the two points on the sphere is given by

\begin{aligned} ‖ 𝝈 (𝑤_{1}) - 𝝈 (𝑤_{2}) ‖ \\ \begin{matrix}  \end{matrix} = \sqrt{\begin{aligned} {(\frac{2 ℜ𝔢 𝑤_{1}}{{| 𝑤_{1} |}^{2} + 1} - \frac{2 ℜ𝔢 𝑤_{2}}{{| 𝑤_{2} |}^{2} + 1})}^{2} + {(\frac{2 ℑ𝔪 𝑤_{1}}{{| 𝑤_{1} |}^{2} + 1} - \frac{2 ℑ𝔪 𝑤_{2}}{{| 𝑤_{2} |}^{2} + 1})}^{2} \\ \begin{matrix}  \end{matrix} + {(\frac{{| 𝑤_{1} |}^{2} - 1}{{| 𝑤_{1} |}^{2} + 1} - \frac{{| 𝑤_{2} |}^{2} - 1}{{| 𝑤_{2} |}^{2} + 1})}^{2} \end{aligned}} \\ \begin{matrix}  \end{matrix} = \frac{1}{({| 𝑤_{1} |}^{2} + 1) ({| 𝑤_{2} |}^{2} + 1)} \\ \begin{matrix}  \end{matrix} \times \sqrt{\begin{aligned} 4 {ℜ𝔢 [𝑤_{1} ({| 𝑤_{2} |}^{2} + 1) - 𝑤_{2} ({| 𝑤_{1} |}^{2} + 1)]}^{2} \\ \begin{matrix}  \end{matrix} + 4 {ℑ𝔪 [𝑤_{1} ({| 𝑤_{2} |}^{2} + 1) - 𝑤_{2} ({| 𝑤_{1} |}^{2} + 1)]}^{2} \\ \begin{matrix}  \end{matrix} + {[({| 𝑤_{1} |}^{2} - 1) ({| 𝑤_{2} |}^{2} + 1) - ({| 𝑤_{2} |}^{2} - 1) ({| 𝑤_{1} |}^{2} + 1)]}^{2} \end{aligned}} \\ \begin{matrix}  \end{matrix} = \frac{2}{𝜌_{1} 𝜌_{2}} \sqrt{{| 𝑤_{1} 𝜌_{2} - 𝑤_{2} 𝜌_{1} |}^{2} + {(𝜌_{1} - 𝜌_{2})}^{2}} (where 𝜌_{𝑖} = {| 𝑤_{𝑖} |}^{2} + 1) \\ \begin{matrix}  \end{matrix} = \frac{2}{𝜌_{1} 𝜌_{2}} \sqrt{𝜌_{2}^{2} (𝜌_{1} - 1) + 𝜌_{1}^{2} (𝜌_{2} - 1) - 2 𝜌_{2} 𝜌_{1} ℜ𝔢 [𝑤_{1} \overline{𝑤_{2}}] + {(𝜌_{1} - 𝜌_{2})}^{2}} \\ \begin{matrix}  \end{matrix} = \frac{2}{𝜌_{1} 𝜌_{2}} \sqrt{𝜌_{2}^{2} 𝜌_{1} + 𝜌_{1}^{2} 𝜌_{2} - 2 𝜌_{2} 𝜌_{1} ℜ𝔢 [𝑤_{1} \overline{𝑤_{2}}] - 2 𝜌_{1} 𝜌_{2}} \\ \begin{matrix}  \end{matrix} = \frac{2}{\sqrt{𝜌_{1} 𝜌_{2}}} \sqrt{𝜌_{2} + 𝜌_{1} - 2 ℜ𝔢 [𝑤_{1} \overline{𝑤_{2}}] - 2} = \frac{2 | 𝑤_{1} - 𝑤_{2} |}{\sqrt{𝜌_{1} 𝜌_{2}}} = 𝑘 (𝑤_{1}, 𝑤_{2}) \end{aligned}

Thus, $𝑘$ measures the chordal distance between points on the Riemann sphere, and is thus geometrically invariant under rotations of the sphere. The mapping $𝑊$ of $𝑤$ is precisely such a transformation: geometrically it rotates points on the Riemann sphere. Geometrically, the invariance of $𝑚_{0}$ is now clear.

Let $𝑤 = 𝑢 + i 𝑣$ be a point in the projected plane and let $d 𝑢 d 𝑣$ be an area element on the plane. Then the corresponding area element on the sphere is equal to:

\begin{aligned} d 𝐴_{𝑆} & = ‖ 𝝈_{𝑢}^{'} \times 𝝈_{𝑣}^{'} ‖ d 𝑢 d 𝑣 \\ = \frac{d 𝑢 d 𝑣}{{({| 𝑤 |}^{2} + 1)}^{4}} ‖ (\begin{matrix} 2 𝑣^{2} - 2 𝑢^{2} + 2 \\ - 4 𝑣 𝑢 \\ 4 𝑢 \end{matrix}) \times (\begin{matrix} - 4 𝑢 𝑣 \\ 2 𝑢^{2} - 2 𝑣^{2} + 2 \\ 4 𝑣 \end{matrix}) ‖ \\ = \frac{4 d 𝑢 d 𝑣}{{({| 𝑤 |}^{2} + 1)}^{4}} ‖ (\begin{matrix} - 2 𝑢 𝑣^{2} - 2 𝑢^{3} - 2 𝑢 \\ - 2 𝑢^{2} 𝑣 - 2 𝑣^{3} - 2 𝑣 \\ 1 - {(𝑢^{2} + 𝑣^{2})}^{2} \end{matrix}) ‖ = \frac{4 d 𝑢 d 𝑣}{{({| 𝑤 |}^{2} + 1)}^{3}} ‖ (\begin{matrix} - 2 𝑢 \\ - 2 𝑣 \\ 1 - {| 𝑤 |}^{2} \end{matrix}) ‖ \\ = \frac{4 d 𝑢 d 𝑣}{{({| 𝑤 |}^{2} + 1)}^{3}} \sqrt{2 {| 𝑤 |}^{2} + 1 + {| 𝑤 |}^{4}} = \frac{4 d 𝑢 d 𝑣}{{({| 𝑤 |}^{2} + 1)}^{2}} \end{aligned}

The surface area of a surface $𝐸 \subseteq 𝑆$ is given by

\int_{𝐸} d 𝐴_{𝑆} = \iint_{𝝈^{- 1} (𝐸)} \frac{4 d 𝑢 d 𝑣}{{({| 𝑤 |}^{2} + 1)}^{2}} = - 2 i \iint_{𝝈^{- 1} (𝐸)} \frac{d \overline{𝑤} d 𝑤}{{({| 𝑤 |}^{2} + 1)}^{2}} .

For $𝑤 = 𝑓 (𝑧)$ , $𝐸 = 𝝈 \circ 𝑓 (𝐷 (0, 𝑟))$ , pulling back to $𝑧$ gives

\begin{aligned} \frac{1}{4 π} \iint_{𝝈 (𝑓 (𝐷 (0, 𝑟)))} d 𝐴_{𝑆} & = \frac{1}{2 π i} \iint_{𝐷 (0, 𝑟)} \frac{(\overline{𝑓^{'} (𝑧)} d \overline{𝑧}) \land (𝑓^{'} (𝑧) d 𝑧)}{{({| 𝑓 (𝑧) |}^{2} + 1)}^{2}} \\ = \frac{1}{4 π} \iint_{𝐷 (0, 𝑟)} 𝑓^{♯} {(𝑧)}^{2} d 𝑥 d 𝑦 \\ = \frac{[area of 𝑓 \circ 𝐷 (0, 𝑟) on 𝑆^{2} with multiplicities]}{[area of 𝑆^{2} (Riemann sphere)]} \\ = 𝐴 (𝑟, 𝑓) (☆) \end{aligned}

or the area of the image of $𝑓$ on the Riemann sphere, counted according to multiplicities (referring to overlaps when not univalent), divided by the area of the Riemann sphere ( $4 π$ ). In some sense, $𝐴$ measures the amount by which the image of $𝑓$ covers the Riemann sphere.

(Observe the similarity between the first integral of $(☆)$ and the Lusin Area Theorem (Theorem 3.1.1 $Theorem 3.1.1 Theorem 3.1.1$ )!)

This completes the geometric explanation for the invariances of $(a)$ $Part 3 Part 3$ , $(b)$ $Part 3 Part 3$ , and $(c)$ $Part 3 Part 3$ in the proof of Theorem 14.3.5 $Theorem 14.3.5 Theorem 14.3.5$ . The covered area and chordal distance are invariant under rotations of $𝑆$ . In particular, we rotate the spherical point of $𝑎$ to the north pole $(0, 0, 1)$ corresponding to $\infty$ .

Remark.

It is noteworthy that the original Ahlfors–Shimizu formulation uses the Riemann sphere centered at $(0, 0, 1 / 2)$ with diameter $1$ , which yields a slightly different proximity function (without the $/ 2$ factor) and spherical derivative (without the factor of 2). (Note that the characteristic does not change as the amount a function covers the extended plane (or either sphere) is independent on the size of this sphere).

This normalization is present throughout Ahlfors’ works; in Section 8 $Section 8 Section 8$ , the spherical and hyperbolic metrics are also defined without a $2 \cdot$ factor (and hence yield curvatures of $\pm 4$ ). There is little fundamental difference between the two normalizations, and we have adopted the sphere of diameter $2$ to uniformize better with the previous sections.