Conformal maps and the Schwarz lemma

For a holomorphic f near z₀ with $f'(z_0)\ne0$, the first-order approximation is multiplication by that nonzero complex number. Two smooth curves meeting with nonzero tangent vectors therefore have their oriented angle preserved after mapping. The real Jacobian determinant equals $|f'(z_0)|^2>0$, so orientation is preserved and the inverse function theorem supplies a local inverse. These are local conclusions at a regular point, with no promise of global one-to-one behavior.

The exponential has nonzero derivative everywhere but repeats values after a vertical displacement of 2πi, so it is not injective on the whole plane. The map z² has zero derivative at the origin and doubles arguments there, while being locally conformal at every nonzero point. A conformal equivalence between domains requires a bijective holomorphic map with a holomorphic inverse. Specifying the domains is as necessary as writing an attractive formula for the transformation.

Let f map the open unit disk holomorphically into itself and satisfy f(0)=0. Divide by z and fill the removable value at zero with f′(0). On a smaller circle of radius r, the quotient's modulus is at most 1/r. The maximum modulus principle bounds it on that smaller disk. Letting r tend to one shows that the quotient has modulus at most one everywhere, yielding the radial and derivative inequalities of the Schwarz lemma.

If equality in the radial inequality holds at a nonzero interior point, the quotient attains its maximum modulus inside the disk and must be a constant of modulus one. Equality in the derivative bound gives the same result. Thus f is a rotation of z. Equality at the center in the radial inequality is automatic and gives no rigidity by itself. The normalization f(0)=0 is also essential; general disk automorphisms moving the center require a transformed inequality instead.

Linear fractional transformations are useful because their inverse can be solved algebraically and they map generalized circles to generalized circles. To prove that one maps the upper half-plane to the disk, compare the squared distances to i and −i. For z=x+iy with y positive, the distance to i is smaller than the distance to −i. The corresponding ratio has modulus less than one. This inequality proves the image inclusion, while the explicit inverse proves surjectivity and injectivity.

Such maps can transfer harmonic boundary problems or simplify a contour, but boundary points and poles must be tracked. A pole outside the source domain is harmless for holomorphicity inside it; a missing boundary point may correspond to infinity. Angle preservation concerns interior regular points and does not automatically give every boundary regularity statement. Domain, inverse, derivative, and boundary correspondence are separate checks that together turn a formal substitution into a usable conformal equivalence.