Grunsky matrix

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 . 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
is a non-vanishing univalent function on |"z"| > 1 having a simple pole at ∞ with residue 1:
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
It is a symmetric matrix. Its entries are called the Grunsky coefficients of "g".
Note that
so that the coefficients can be expressed directly in terms of "f". Indeed, if
then for "m", "n" > 0
and "d"0"n" = "d""n"0 is given by
with
Grunsky inequalities.
If "f" is a holomorphic function on the unit disk with Grunsky matrix ("c""nm"), the Grunsky inequalities state that
of a single function "g" or a pair of functions "F", "g" is unitary if and only if the complement of the image of "g" or the union of the images of "F" and "g" has Lebesgue measure zero. So, roughly speaking, in the case of one function the image is a slit region in the complex plane; and in the case of two functions the two regions are separated by a closed Jordan curve.
In fact the infinite matrix "A" acting on the Hilbert space of square summable sequences satisfies
But if "J" denotes complex conjugation of a sequence, then
since "A" is symmetric. Hence
so that "A" is unitary.
Equivalent forms of Grunsky inequalities.
Goluzin inequalities.
If "g"("z") is a normalized univalent function in |"z"| > 1, "z"1, ..., "z""N" are distinct points with |"zn"| > 1 and α1, ..., α"N" are complex numbers, the Goluzin inequalities, proved in 1947 by the Russian mathematician Gennadi Mikhailovich Goluzin (1906-1953), state that
To deduce them from the Grunsky inequalities, let
for "k" > 0.
Conversely the Grunsky inequalities follow from the Goluzin inequalities by taking
where
with "r" > 1, tending to ∞.
Bergman–Schiffer inequalities.
 gave another derivation of the Grunsky inequalities using reproducing kernels and singular integral operators in geometric function theory; a more recent related approach can be found in .
Let "f"("z") be a normalized univalent function in |"z"| < 1, let "z"1, ..., "z""N" be distinct points with |"z""n"| < 1 and let α1, ..., α"N" be complex numbers. The Bergman-Schiffer inequalities state that
To deduce these inequalities from the Grunsky inequalities, set
for "k" > 0.
Conversely the Grunsky inequalities follow from the Bergman-Schiffer inequalities by taking
where
with "r" < 1, tending to 0.
Applications.
The Grunsky inequalities imply many inequalities for univalent functions. They were also used by Schiffer and Charzynski in 1960 to give a completely elementary proof of the Bieberbach conjecture for the fourth coefficient; a far more complicated proof had previously been found by Schiffer and Garabedian in 1955. In 1968 Pedersen and Ozawa independently used the Grunsky inequalities to prove the conjecture for the sixth coefficient.
In the proof of Schiffer and Charzynski, if
is a normalized univalent function in |"z"| < 1, then
is an odd univalent function in |"z"| > 1.
Combining Gronwall's area theorem for "f" with the Grunsky inequalities for the first 2 x 2 minor of the Grunsky matrix of "g" leads to a bound for |"a"4| in terms of a simple function of "a"2 and a free complex parameter. The free parameter can be chosen so that the bound becomes a function of half the modulus of "a"2 and it can then be checked directly that this function is no greater than 4 on the range [0,1].
As Milin showed, the Grunsky inequalities can be exponentiated. The simplest case proceeds by writing
with "a""n"("w") holomorphic in |"w"| < 1.
The Grunsky inequalities, with λ"n" = "w""n" imply that
On the other hand, if
as formal power series, then the first of the Lebedev–Milin inequalities (1965) states that
Equivalently the inequality states that if "g"("z") is a polynomial with "g"(0) = 0, then
where "A" is the area of "g"("D"),
To prove the inequality, note that the coefficients are determined by the recursive formula
so that by the Cauchy–Schwarz inequality
The quantities "c""n" obtained by imposing equality here:
satisfy formula_31 and hence, reversing the steps,
In particular defining "b""n"("w") by the identity
the following inequality must hold for |"w"| < 1
Beurling transform.
The Beurling transform (also called the Beurling-Ahlfors transform and the Hilbert transform in the complex plane) provides one of the most direct methods of proving the Grunsky inequalities, following and .
The Beurling transform is defined on "L"2(C) as the operation of multiplication by formula_35 on Fourier transforms. It thus defines a unitary operator. It can also be defined directly as a principal value integral
For any bounded open region Ω in C it defines a bounded operator "T"Ω from the conjugate of the Bergman space of Ω onto the Bergman space of Ω: a square integrable holomorphic function is extended to 0 off Ω to produce a function in "L"2(C) to which "T" is applied and the result restricted to Ω, where it is holomorphic. If "f" is a holomorphic univalent map from the unit disk "D" onto Ω then the Bergman space of Ω and its conjugate can be identified with that of "D" and "T"Ω becomes the singular integral operator with kernel
It defines a contraction. On the other hand, it can be checked that "T""D" = 0 by computing directly on powers formula_38 using Stokes theorem to transfer the integral to the boundary.
It follows that the operator with kernel
acts as a contraction on the conjugate of the Bergman space of "D". Hence, if
then
Grunsky operator and Fredholm determinant.
If Ω is a bounded domain in C with smooth boundary, the operator "T"Ω can be regarded as a bounded antilinear contractive operator on the Bergman space "H" = "A"2(Ω). It is given by the formula
for "u" in the Hilbert space "H"= "A"2(Ω). "T"Ω is called the Grunsky operator of Ω (or "f"). Its realization on "D" using a univalent function "f" mapping "D" onto Ω and the fact that "TD" = 0 shows that it is given by restriction of the kernel
and is therefore a Hilbert–Schmidt operator.
The antilinear operator "T" = "T"Ω satisfies the self-adjointness relation
for "u", "v" in "H".
Thus "A" = "T"2 is a compact self-adjont linear operator on "H" with
so that "A" is a positive operator. By the spectral theorem for compact self-adjoint operators, there is an orthonormal basis "u""n" of "H" consisting of eigenvectors of "A":
where μ"n" is non-negative by the positivity of "A". Hence
with λ"n" ≥ 0. Since "T" commutes with "A", it leaves its eigenspaces invariant. The positivity relation shows that it acts trivially on the zero eigenspace. The other non-zero eigenspaces are all finite-dimensional and mutually orthogonal. Thus an orthonormal basis can be chosen on each eigenspace so that:
The non-zero λ"n" (or sometimes their reciprocals) are called the Fredholm eigenvalues of Ω:
If Ω is a bounded domain that is not a disk, Ahlfors showed that
The Fredholm determinant for the domain Ω is defined by
Note that this makes sense because "A" = "T"2 is a trace class operator.
 showed that if formula_53 and "f" fixes 0, then
Here the norms are in the Bergman spaces of "D" and its complement "D""c" and "g" is a univalent map from "D""c" onto Ω"c" fixing ∞.
A similar formula applies in the case of a pair of univalent functions (see below).
Singular integral operators on a closed curve.
Let Ω be a bounded simply connected domain in C with smooth boundary "C" = ∂Ω. Thus there is a univalent holomorphic map "f" from the unit disk "D" onto Ω extending to a smooth map between the boundaries "S"1 and "C".