---
tag: XKRR37N916
kind: concept
name: Grunsky matrix
msc: [30C50, 30C55]
sources:
  - url: https://en.wikipedia.org/wiki/Grunsky_matrix
same_as:
  wikidata: Q5612135
  wikipedia: 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|BYZDAZNHP4]] and were a main tool in the work on coefficient problems before the proof of the [[Bieberbach conjecture|ZMRFSYVWQT]].

## Explanations

### L1 human

A univalent function $g$ never takes the same value twice, so the difference quotient $(g(z) - g(\zeta))/(z - \zeta)$ never vanishes.
Its logarithm is then a nice function of two variables, which can be expanded in powers of $1/z$ and $1/\zeta$.
The Grunsky matrix lists the coefficients of this expansion.
For $g(z) = z + 1/z$ the expansion is $\log(1 - 1/(z\zeta))$, and the matrix is diagonal with entries $1, 1/2, 1/3, \ldots$.