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Mathematics LibreTexts

18.10: Schwarz-Pick theorem

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    58643
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    The following theorem shows that the metric in the conformal disc model naturally appears in other branches of mathematics. We do not give its proof, but it can be found in any textbook on geometric complex analysis.

    Suppose that \(\mathbb{D}\) denotes the unit disc in the complex plane centered at \(0\); that is, a complex number \(z\) belongs to \(\mathbb{D}\) if and only if \(|z|<1\).

    Let us use the disc \(\mathbb{D}\) as a h-plane in the conformal disc model; the h-distance between \(z, w\in\mathbb{D}\) will be denoted by \(d_h(z,w)\); that is,

    \(d_h(z,w) := ZW_h,\)

    where \(Z\) and \(W\) are h-points with complex coordinates \(z\) and \(w\) respectively.

    A function \(f:\mathbb{D}\to \mathbb{C}\) is called holomorphic if for every \(z\in \mathbb{D}\) there is a complex number \(s\) such that

    \(f(z+w)=f(z)+s\cdot w+o(|w|).\)

    In other words, \(f\) is complex-differentiable at any \(z\in\mathbb{D}\). The complex number \(s\) is called the derivative of \(f\) at \(z\), or briefly \(s=f'(z)\).

    Theorem \(\PageIndex{1}\) Schwarz-Pick theorem

    Schwarz–Pick theorem Assume \(f\: \mathbb{D}\to \mathbb{D}\) is a holomorphic function. Then

    \(d_h(f(z),f(w))\le d_h(z,w)\)

    for any \(z,w\in \mathbb{D}\).

    If the equality holds for one pair of distinct numbers \(z,w\in \mathbb{D}\), then it holds for any pair. In this case \(f\) is a fractional linear transformation as well as a motion of the h-plane.

    Exercise \(\PageIndex{1}\)

    Show that if a fractional linear transformation \(f\) appears in the equality case of Schwarz–Pick theorem, then it can be written as

    \(f(z)=\dfrac{v\cdot z+\bar w}{w\cdot z+\bar v}.\)

    where \(v\) and \(w\) are complex constants such that \(|v|>|w|\).

    Hint

    Note that \(f = \dfrac{a \cdot z + b}{c \cdot z + d}\) preserves the unit circle \(|z| = 1\). Use Corollary 10.6.1 and Proposition 18.12 to show that \(f\) commutes with the inversion \(z \mapsto 1/\bar{z}\). In other words, \(1/f(z) = f(1/\bar{z})\) or

    \(\dfrac{\bar{c} \cdot \bar{z} + \bar{d}}{\bar{a} \cdot \bar{z} + \bar{b}} = \dfrac{a/\bar{z} + b}{c/\bar{z} + d}\)

    for any \(z \in \bar{\mathbb{C}}\). The latter identity leads to the required statements. The condition \(|w| < |v|\) follows since \(f(0) \in \mathbb{D}\).

    Exercise \(\PageIndex{2}\)

    Recall that hyperbolic tangent \(\tanh\) is defined on page . Show that

    \(\tanh [\tfrac12\cdot d_h(z,w)]=\left|\frac{z-w}{1-z\cdot\bar w}\right|.\)

    Conclude that the inequality in Schwarz–Pick theorem can be rewritten as

    \(\left|\frac{z'-w'}{1-z'\cdot\bar w'}\right|\le\left|\frac{z-w}{1-z\cdot\bar w}\right|,\)

    where \(z'=f(z)\) and \(w'=f(w)\).

    Hint

    Note that the inverses of the points \(z\) and \(w\) have complex coordinates \(1/\bar{z}\) and \(1/\bar{w}\). Apply Exercise 12.9.2 and simplify.

    The second part follows since the function \(x \mapsto \tanh (\dfrac{1}{2} \cdot x)\) is increasing.

    Exercise \(\PageIndex{3}\)

    Show that the Schwarz lemma stated below follows from Schwarz–Pick theorem.

    Hint

    Apply Schwarz-Pick theorem for a function \(f\) such that \(f(0) = 0\) and then apply Lemma 12.3.2.

    Lemma \(\PageIndex{1}\) Schwarz lemma

    Let \(f\: \mathbb{D}\to \mathbb{D}\) be a holomorphic function and \(f(0)=0\). Then \(|f(z)|\le |z|\) for any \(z\in \mathbb{D}\).

    Moreover, if equality holds for some \(z\ne 0\), then there is a unit complex number \(u\) such that \(f(z)=u\cdot z\) for any \(z\in\mathbb{D}\).