the growth theorem

If a univalent map of the disk has its scale fixed at the centre, how far out and how far in can it carry a point that lies a fraction r of the way to the boundary? You might fear there is no control — a one-to-one map can stretch a lot. The growth theorem says, reassuringly, that there is tight two-sided control: at each radius the modulus |f(z)| is trapped between an explicit smallest and largest value, and both extremes are realized by the Koebe function.

Precisely, for f in the class S and a point z with |z| = r < 1, the growth theorem states r/(1 + r)^2 <= |f(z)| <= r/(1 - r)^2. The upper bound says f cannot blow up faster than the Koebe rate; the lower bound says f cannot collapse toward 0 faster than the Koebe rate either. Both bounds come from integrating the distortion theorem (which bounds |f'|) along a radius, and both are sharp — equality on each side is achieved by suitable rotations of the Koebe function k(z) = z/(1 - z)^2, whose modulus on the real axis is exactly r/(1 - r)^2.

This is what makes the class S a genuinely controlled family rather than a wild zoo: knowing only that a map is normalized and univalent pins down its size at every interior radius to within a fixed envelope. Letting r approach 1 in the lower bound also recovers the Koebe one-quarter theorem (the omitted values are at least 1/4 out). A caution: the bounds are about the MODULUS |f(z)| at radius r, not about which direction f(z) points; and they apply only to the normalized class S — drop f'(0) = 1 and the constants must be rescaled accordingly.