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The Poincare distance has the formula $2 artanh ∣ φ_{z} (w) ∣$ and is disc-automorphism invariant

Statement

For $z, w \in D$ , the Poincare distance on the unit disc satisfies

$d_{D} (z, w) = 2 artanh ∣ φ_{z} (w) ∣,$

where $φ_{z}$ is the Blaschke factor carrying $z$ to $0$ . Moreover every disc automorphism preserves this distance.

Facts & Assumptions

Given: The Poincare metric and distance on $D$ .

[F1]

The Poincare length and distance are defined by integrating $2 ∣ d z ∣ / (1 - ∣ z ∣^{2})$ along piecewise $C^{1}$ curves (The Poincare metric and distance on the unit disc).

[F2]

Every Blaschke factor is an automorphism of $D$ (Blaschke factors are automorphisms of the disc).

[F3]

Every automorphism of $D$ is a rotated Blaschke factor (Every automorphism of the disc is a rotated Blaschke factor).

Proof

technique · direct

1.1

For a Blaschke factor $φ_{a}$ , a direct differentiation gives $φ_{a}^{'} (z) = - (1 - ∣ a ∣^{2}) / (1 - \overline{a} z)^{2}$ and $1 - ∣ φ_{a} (z) ∣^{2} = (1 - ∣ a ∣^{2}) (1 - ∣ z ∣^{2}) / ∣ 1 - \overline{a} z ∣^{2}$ . Therefore $2 ∣ φ_{a}^{'} (z) ∣ / (1 - ∣ φ_{a} (z) ∣^{2}) = 2 / (1 - ∣ z ∣^{2})$ , so $φ_{a}$ preserves Poincare length of every piecewise $C^{1}$ curve.

F1F2givenalgebra

2.1

Since length is preserved under $φ_{a}$ , taking infima in [F1] gives $d_{D} (φ_{a} (z), φ_{a} (w)) = d_{D} (z, w)$ . By [F3], every disc automorphism is a composition of a Blaschke factor and a rotation, and rotations satisfy the same identity, so every disc automorphism preserves $d_{D}$ .

F1F2F3step 1.1algebra

3.1

By step 2.1, $d_{D} (z, w) = d_{D} (0, φ_{z} (w))$ . Write $r = ∣ φ_{z} (w) ∣$ . If $r = 0$ , then $φ_{z} (w) = 0$ , so $z = w$ and both sides are $0 = 2 artanh 0$ . If $r > 0$ , the radial segment $γ (t) = t φ_{z} (w) / r$ from $0$ to $φ_{z} (w)$ has Poincare length $\int_{0}^{r} 2 d t / (1 - t^{2}) = 2 artanh r$ , so $d_{D} (0, φ_{z} (w)) \leq 2 artanh r$ .

F1step 2.1algebra

4.1

For any piecewise $C^{1}$ curve $γ$ from $0$ to $φ_{z} (w)$ , one has $∣ (∣ γ ∣)^{'} ∣ \leq ∣ γ^{'} ∣$ , hence $ℓ_{D} (γ) \geq \int 2 ∣ (∣ γ ∣)^{'} ∣ / (1 - ∣ γ ∣^{2}) d t \geq \int_{0}^{r} 2 d s / (1 - s^{2}) = 2 artanh r$ . Taking the infimum over all such curves gives the reverse inequality.

F1step 3.1algebra

5.1

Combining steps 3.1 and 4.1 yields $d_{D} (z, w) = 2 artanh ∣ φ_{z} (w) ∣$ , and step 2.1 gives automorphism invariance.

step 2.1step 3.1step 4.1∎

Depends on

Used by

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Sources

Jiri Lebl, Guide to Cultivating Complex Analysis, §3.5 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §2 (standard reference, not scraped)

The Poincare distance has the formula 2artanh⁡∣φz(w)∣2artanh∣φz​(w)∣ and is disc-automorphism invariant