the Koebe one-quarter theorem
/ KUR-buh /
Suppose you take the unit disk and bend it with any one-to-one holomorphic map that fixes the centre and keeps the local scale equal to 1 there. The image can be wildly stretched and reshaped, but it cannot crowd in arbitrarily close to its own centre point: it is forced to contain a whole little disk of a guaranteed radius around the origin. That guaranteed radius is exactly one quarter. This is the Koebe one-quarter theorem, a striking 'you cannot avoid covering this much' statement.
Precisely: if f is in the class S — holomorphic and univalent on the unit disk with f(0) = 0 and f'(0) = 1 — then the image f(disk) contains the open disk of radius 1/4 centred at 0, that is { w : |w| < 1/4 } is a subset of f(disk). Equivalently, f omits no value w with |w| < 1/4. The proof is short once you have the Bieberbach bound: feed an omitted value w_0 into the standard transform f / (1 - f/w_0), keep it in S, and apply |a_2| <= 2; the inequality rearranges to |w_0| >= 1/4. So a single coefficient bound forces a covering radius.
The constant 1/4 is best possible and the Koebe function is the witness: it omits the slit starting exactly at -1/4, so it covers no disk larger than radius 1/4. The theorem is the cornerstone of the quantitative side of the Riemann mapping theorem — it gives an honest lower bound on how big a normalized conformal image must be, which is what makes constructive and numerical conformal mapping controllable. A common misreading: the covered disk is centred at the IMAGE of 0 (which is 0 by normalization), not at an arbitrary point; and the 1/4 is the radius, not the diameter.
If f in S omitted the value w_0 = 1/5 (with |w_0| = 1/5 < 1/4), the one-quarter theorem says this is impossible — f must hit it. The boundary case is the Koebe function, which omits exactly the ray from -1/4 outward, so the covered disk reaches precisely radius 1/4 and no more.
Every class-S image contains the disk |w| < 1/4; the Koebe slit shows 1/4 cannot be improved.
1/4 is the radius of the GUARANTEED covered disk centred at the origin, not the diameter and not a bound on omitted values larger than 1/4 — outside that disk a univalent map may omit plenty (Koebe omits a whole ray).