the class S of normalized univalent functions

When mathematicians study a whole family of maps, they like to strip away the trivial freedoms first — the overall position, rotation, and scale — so that what remains is the genuine geometric content. The class S is exactly this: the standard, normalized stock of univalent maps of the unit disk. Pinning down the normalization turns a sprawling collection into a clean test bed where sharp inequalities and extremal examples can be compared on equal footing.

Precisely, S is the set of functions f that are holomorphic and univalent (one-to-one) on the open unit disk |z| < 1 and satisfy the two normalizations f(0) = 0 and f'(0) = 1. Because of these, every member has a Taylor series of the form f(z) = z + a_2 z^2 + a_3 z^3 + ... ; the leading term is forced to be exactly z. Any univalent g on the disk can be reduced to a member of S by subtracting g(0) and dividing by g'(0), so studying S loses nothing essential. The whole circle of famous results — the Bieberbach bound |a_2| <= 2, the Koebe one-quarter theorem, the growth and distortion theorems, and the Bieberbach conjecture |a_n| <= n — are statements about this single class.

S is a remarkably rigid family: it is compact in the topology of locally uniform convergence, so extremal problems over S actually attain their extrema, and a single function, the Koebe function, turns out to be the extremal example again and again. A caution about scope: S is the disk version with that specific normalization. Its companion is the class Sigma of functions univalent on the EXTERIOR of the disk, with a different normalization; the area theorem is naturally proved for Sigma first and then transferred to S. Do not conflate the two — they are dual halves of the same theory.