In complex analysis and geometric function theory, the Grunsky matrices, or Grunsky operators, are infinite matrices introduced in 1939 by Helmut Grunsky. The matrices correspond to either a single holomorphic function on the unit disk or a pair of holomorphic functions on the unit disk and its complement. The Grunsky inequalities express boundedness properties of these matrices, which in general are contraction operators or in important special cases unitary operators. As Grunsky showed, these inequalities hold if and only if the holomorphic function is univalent. The inequalities are equivalent to the inequalities of Goluzin, discovered in 1947. Roughly speaking, the Grunsky inequalities give information on the coefficients of the logarithm of a univalent function; later generalizations by Milin, starting from the Lebedev–Milin inequality, succeeded in exponentiating the inequalities to obtain inequalities for the coefficients of the univalent function itself. The Grunsky matrix and its associated inequalities were originally formulated in a more general setting of univalent functions between a region bounded by finitely many sufficiently smooth Jordan curves and its complement: the results of Grunsky, Goluzin and Milin generalize to that case. Historically the inequalities for the disk were used in proving special cases of the Bieberbach conjecture up to the sixth coefficient; the exponentiated inequalities of Milin were used by de Branges in the final solution. A detailed exposition using these methods can be found in Hayman (1994). The Grunsky operators and their Fredholm determinants are also related to spectral properties of bounded domains in the complex plane. The operators have further applications in conformal mapping, Teichmüller theory and conformal field theory.
Grunsky Matrix If f(z) is a holomorphic univalent function on the unit disk, normalized so that f(0) = 0 and f′(0) = 1, the function
g ( z ) = f ( z − 1 ) − 1 {\displaystyle g(z)=f(z^{-1})^{-1}}
is a non-vanishing univalent function on |z| > 1 having a simple pole at ∞ with residue 1:
g ( z ) = z + b 0 + b 1 z − 1 + b 2 z − 2 + ⋯ {\displaystyle g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots }
The same inversion formula applied to g gives back f and establishes a one-one correspondence between these two classes of function. The Grunsky matrix (cnm) of g is defined by the equation
log g ( z ) − g ( ζ ) z − ζ = − ∑ m , n > 0 c n m z − m ζ − n {\displaystyle \log {\frac {g(z)-g(\zeta )}{z-\zeta }}=-\sum _{m,n>0}c_{nm}z^{-m}\zeta ^{-n}}
It is a symmetric matrix. Its entries are called the Grunsky coefficients of g. Note that
log g ( z − 1 ) − g ( ζ − 1 ) z − 1 − ζ − 1 = log f ( z ) − f ( ζ ) z − ζ − log f ( z ) z − log f ( ζ ) ζ , {\displaystyle \log {g(z^{-1})-g(\zeta ^{-1}) \over z^{-1}-\zeta ^{-1}}=\log {f(z)-f(\zeta ) \over z-\zeta }-\log {f(z) \over z}-\log {f(\zeta ) \over \zeta },}
so that the coefficients can be expressed directly in terms of f. Indeed, if
log f ( z ) − f ( ζ ) z − ζ = − ∑ m , n ≥ 0 d m n z n ζ n , {\displaystyle \log {f(z)-f(\zeta ) \over z-\zeta }=-\sum _{m,n\geq 0}d_{mn}z^{n}\zeta ^{n},}
then for m, n > 0
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