Grunsky matrix

Summary

Let $g$ be a univalent function on the outside of the unit disc. Its Grunsky matrix lists the coefficients of the double series of $\log\frac{g(z) - g(\zeta)}{z - \zeta}$. Univalence is reflected in the Grunsky inequalities, which say that a weighted form of this matrix is a contraction on the sequence space $\ell^2$. These inequalities refine the Grönwall area theorem and were a main tool in the work on coefficient problems before the proof of the Bieberbach conjecture.