the Koebe function
/ KUR-buh /
Every good theory has a hero example that keeps showing up at the edge of every inequality. In the theory of univalent functions that hero is the Koebe function. It is the single map of the disk that stretches the boundary as violently as a one-to-one map possibly can, and as a reward it sits exactly on the boundary of nearly every sharp bound in the subject. If you want to know whether an estimate for the class S can be improved, you test it against Koebe.
It is the explicit function k(z) = z/(1 - z)^2, defined on the unit disk. A short computation gives its Taylor series k(z) = z + 2 z^2 + 3 z^3 + ... , so its n-th coefficient is exactly n — the largest the Bieberbach conjecture allows. Geometrically, k maps the open unit disk one-to-one onto the entire plane with a single straight ray removed: the slit from -1/4 going out to minus infinity along the negative real axis (the 'slit plane'). The boundary circle of the disk is wrapped onto that slit, doubled over, with the point z = -1 sent off to infinity. Rotations e^(-i theta) k(e^(i theta) z) point the slit in other directions and form the full family of extremals.
The Koebe function is the universal extremal: it attains equality in |a_2| <= 2 (with a_2 = 2), in the Koebe one-quarter theorem (the omitted slit starts exactly at distance 1/4 from 0), in the growth and distortion theorems, and in every case of the Bieberbach conjecture |a_n| <= n. A caution worth stating: k(z) = z/(1 - z)^2 is univalent on the disk only — it is not one-to-one on any larger disk, since the pole and the doubling at z = 1 spoil it. Its magic is that it pushes the disk to the absolute geometric limit while staying schlicht.
Write k(z) = z/(1 - z)^2 and note that 1/(1 - z)^2 = sum_{n>=1} n z^(n-1) is the derivative of 1/(1 - z). Multiplying by z gives k(z) = sum_{n>=1} n z^n = z + 2 z^2 + 3 z^3 + ... . The image is the plane minus the ray (-infinity, -1/4]; the nearest omitted point to the origin is -1/4.
The Koebe function maps the disk onto the plane minus a radial slit, with coefficients a_n = n.
The omitted slit reaches in to -1/4, not -1/4 to -1; that single distance 1/4 is exactly the covering radius in the Koebe one-quarter theorem. The function is named for Paul Koebe, not 'Koebe' a place.