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Schwarz Lemma
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f(z) = z^n, n > 1, f(0) = 0
|f(z)| <= |z|^n within the unit disk; inequality is strict unless f is a rotation.
f(z) = z^3, f(0) = 0
|f(z)| <= |z|^3 within the unit disk; function is generally a strict contraction.
f(z) = a\frac{z}{2}, |a| < 2, f(0) = 0
|f(z)| <= \frac{1}{2}|z| within the unit disk; f compresses the disk.
f(z) = 0, f(0) = 0
|f(z)| = 0; function maps every point to 0.
f(z) = -\frac{1}{2}z, f(0) = 0
|f(z)| <= \frac{1}{2}|z| within the unit disk; function is a contraction mapping.
f(z) = e^{i\theta}z, \theta \in \mathbb{R}, f(0) = 0
f maps the unit disk to itself rotationally; |f(z)| = |z|.
f(z) = a\sin(z), |a| < 1, f(0) = 0
In general, f does not satisfy Schwarz lemma unless restricted to smaller disk, where ensures .
f(z) = ae^{bz}, |a| < e^{-|b|}, f(0) = 0
f maps the unit disk into itself as long as is sufficiently small.
f(z) = a\frac{z^2}{2}, |a| = 1, f(0) = 0
f is not guaranteed to map the unit disk into itself; may have equality for |z| = 1.
f(z) = \frac{az}{z+1}, |a| < 1, f(0) = 0
f maps unit disk to inside itself except possibly at z = -1, where f has a pole outside the unit disk.
f(z) = \frac{z}{1-|a|z}, |a| < 1, f(0) = 0
f maps the unit disk into a smaller disk of radius \frac{1}{|a|} within the unit disk.
f(z) = iz, f(0) = 0
|f(z)| = |z|; the function represents a 90-degree rotation.
f(z) = az^2, |a| < 1, f(0) = 0
f maps the unit disk into a smaller disk; |f(z)| <= |a||z|^2 within the unit disk.
f(z) = \frac{az}{1+z}, |a| < 1, f(0) = 0
f maps the unit disk strictly within itself.
f(z) = az + bz^2, |a| + |b| < 1, f(0) = 0
|f(z)| <= |z| within the unit disk; function is a sum of linear and quadratic terms.
f(z) = az, |a| < 1, f(0) = 0
|f(z)| <= |z| within the unit disk; also, if |f(z)| = |z| for some non-zero z, then f is a rotation: f(z) = e^{i\theta}z.
f(z) = z + az^3, |a| < \frac{1}{3}, f(0) = 0
f is within unit disk for |z| < 1; |f(z)| is strictly less than |z| due to cubic term.
f(z) = az^4, |a| = 1/4, f(0) = 0
|f(z)| <= \frac{1}{4}|z|^4 within the unit disk.
f(z) = az/(1-z)(1+z), |a| < 0.5, f(0) = 0
f maps unit disk within itself, denominator keeps |f(z)| from growing too large.
f(z) = (az + bz^2)/(1+c|z|^2), |a|+|b|+|c| < 1, f(0) = 0
f maps unit disk within itself, given the conditions on a, b, and c.
f(z) = a\frac{z}{1-|b|z^2}, |a|*|b| < 1, f(0) = 0
f maps unit disk to a subset of itself due to the denominator constraining |f(z)|.
f(z) = 0.5z(2 - |z|), f(0) = 0
|f(z)| <= |z| within the unit disk.
f(z) = z - z^2, f(0) = 0
|f(z)| <= |z| within the unit disk; quadratic term acts to reduce the linear growth.
f(z) = az(1-z), |a| < 1, f(0) = 0
f maps unit disk to inside itself, |1-z| remains bounded with |z|<1.
f(z) = az(1 - b|z|^2), |a| < 1/(1+b), f(0) = 0
f maps unit disk into itself; growth term moderated by |z|^2.
f(z) = z - z^3, f(0) = 0
|f(z)| <= |z| within the unit disk; third power term reduces the linear term's impact.
f(z) = ae^{bz^2}, |a| < e^{-b}, f(0) = 0
f maps the unit disk into itself, as the exponential growth due to z^2 is bounded by a factor |a|.
f(z) = cos(az), |a| < 1, f(0) = 1
f does not satisfy the Schwarz lemma since f(0) != 0; however, if centered, it has similar contractive properties.
f(z) = az/(1+z^2), |a| < 0.5, f(0) = 0
f maps the unit disk into itself, |z^2|<1 ensures denominator doesn't reduce the mapping range.
f(z) = a\frac{z-z^2}{1-z}, |a| < 1, f(0) = 0
f maps unit disk within itself, the numerator grows slower than the denominator for |z| < 1.
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