the area theorem

Here is a piece of common sense turned into a sharp inequality: the area of a region cannot be negative. That harmless observation, applied cleverly to the image of a univalent map, becomes the first real engine of the whole theory. The area theorem squeezes the coefficients of a univalent function by demanding that a certain area come out non-negative, and from it tumble the famous bounds for the class S.

The natural setting is the class Sigma: functions g(w) = w + b_0 + b_1/w + b_2/w^2 + ... that are univalent on the EXTERIOR of the unit disk |w| > 1 (with a simple pole at infinity). Such a g sends the exterior onto the complement of some compact set, and the omitted compact set has an area. Computing that area by a contour integral and insisting it be at least zero gives Gronwall's area theorem: the sum over n >= 1 of n |b_n|^2 is at most 1. In words, the tail coefficients cannot all be large at once; their weighted energy is capped by the single number 1. Equality holds only when g maps to the complement of a straight slit.

Why this matters: the area theorem is the seed from which the second-coefficient bound |a_2| <= 2 for the class S grows (apply it to the square-root transform of a class-S function), and that bound in turn powers the Koebe one-quarter theorem and the distortion theorems. So a one-line fact about areas being non-negative ultimately controls how fast univalent functions can grow. The honest caveat: the theorem is stated for Sigma, the exterior class; transferring it to the disk class S takes a clever substitution, not a direct reading.