Schwarz’s lemma: self-maps of the disk fixing 0 cannot expand

If \(f\) maps the unit disk into itself with \(f(0) = 0\), then \(|f(z)| \le |z|\) and \(|f'(0)| \le 1\), with equality only for rotations \(f(z) = e^{i\theta}z\) (Schwarz 1869; Carathéodory gave the general statement in 1912). Blaschke products, the model self-maps of the disk, obey it with room to spare.

import matplotlib.pyplot as plt
import numpy as np

from mathematicskit.complex_analysis import complex_derivative, complex_grid, mobius_transform


def blaschke_factor(a):
    """The disk automorphism (z - a) / (1 - conj(a) z), a Möbius transformation."""
    return lambda z: mobius_transform(z, 1, -a, -np.conj(a), 1)


maps = {
    "rotation e^{0.7i} z": lambda z: np.exp(0.7j) * z,
    "z^2": lambda z: z**2,
    "z (z - 0.5)/(1 - 0.5 z)": lambda z: z * blaschke_factor(0.5)(z),
    "z (z - 0.3i)/(1 + 0.3i z)": lambda z: z * blaschke_factor(0.3j)(z),
}

\(|f(z)| \le |z|\) everywhere in the disk, \(|f'(0)| \le 1\)

z = complex_grid((-0.99, 0.99), (-0.99, 0.99), 201)
z = z[(np.abs(z) < 0.99) & (z != 0)]
fig, ax = plt.subplots()
for name, f in maps.items():
    ratio = np.abs(f(z)) / np.abs(z)
    print(f"{name:26s} max |f(z)|/|z| = {ratio.max():.6f},  |f'(0)| = {abs(complex_derivative(f, 0.0)):.6f}")
    order = np.argsort(np.abs(z))
    ax.plot(np.abs(z)[order][::50], ratio[order][::50], ".", ms=3, label=name)
ax.axhline(1, color="k", lw=1)
ax.set_xlabel("|z|")
ax.set_ylabel("|f(z)| / |z|")
ax.set_title("Schwarz's lemma: the ratio never exceeds 1")
ax.legend(fontsize=8)

rotation e^{0.7i} z        max |f(z)|/|z| = 1.000000,  |f'(0)| = 1.000000
z^2                        max |f(z)|/|z| = 0.990000,  |f'(0)| = 0.000000
z (z - 0.5)/(1 - 0.5 z)    max |f(z)|/|z| = 0.996591,  |f'(0)| = 0.500000
z (z - 0.3i)/(1 + 0.3i z)  max |f(z)|/|z| = 0.994503,  |f'(0)| = 0.300000

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