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

${\displaystyle g(z)=f(z^{-1})^{-1}}$

is a non-vanishing univalent function on |z| > 1 having a simple pole at ∞ with residue 1:

${\displaystyle g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots }$

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 (c_nm) of g is defined by the equation

${\displaystyle \log {\frac {g(z)-g(\zeta )}{z-\zeta }}=-\sum _{m,n>0}c_{nm}z^{-m}\zeta ^{-n}}$

It is a symmetric matrix. Its entries are called the Grunsky coefficients of g.

Note that

${\displaystyle \log {g(z^{-1})-g(\zeta ^{-1}) \over z^{-1}-\zeta ^{-1}}=\log {f(z)-f(\zeta ) \over z-\zeta }-\log {f(z) \over z}-\log {f(\zeta ) \over \zeta },}$

so that the coefficients can be expressed directly in terms of f. Indeed, if

${\displaystyle \log {f(z)-f(\zeta ) \over z-\zeta }=-\sum _{m,n\geq 0}d_{mn}z^{n}\zeta ^{n},}$

then for m, n > 0

${\displaystyle d_{mn}=c_{mn}}$

and d_0n = d_n0 is given by

${\displaystyle \log {\frac {f(z)}{z}}=\sum _{n>0}d_{0n}z^{n}}$

with

${\displaystyle d_{00}=0.}$

If f is a holomorphic function on the unit disk with Grunsky matrix (c_nm), the Grunsky inequalities state that

${\displaystyle \left|\sum _{1\leq m,n\leq N}c_{mn}\lambda _{m}\lambda _{n}\right|\leq \sum _{1\leq n\leq N}{\frac {|\lambda _{n}|^{2}}{n}}}$

for any finite sequence of complex numbers λ₁, ..., λ_N.

The Grunsky coefficients of a normalized univalent function in |z| > 1

${\displaystyle g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots }$

are polynomials in the coefficients b_i which can be computed recursively in terms of the Faber polynomials Φ_n, a monic polynomial of degree n depending on g.

Taking the derivative in z of the defining relation of the Grunsky coefficients and multiplying by z gives

${\displaystyle {\frac {zg'(z)}{g(z)-g(\zeta )}}-{\frac {z}{z-\zeta }}=\sum _{m,n>0}mc_{mn}z^{-m}\zeta ^{-n}.}$

The Faber polynomials are defined by the relation

${\displaystyle {\frac {zg'(z)}{g(z)-w}}=\sum _{n\geq 0}\Phi _{n}(w)z^{-n}.}$

Dividing this relation by z and integrating between z and ∞ gives

${\displaystyle \log {\frac {g(z)-w}{z}}=-\sum _{n\geq 1}{1 \over n}\Phi _{n}(w)z^{-n}.}$

This gives the recurrence relations for n > 0

${\displaystyle \Phi _{n}(w)=(w-b_{0})\Phi _{n-1}(w)-nb_{n}-\sum _{0\leq i\leq n-1}b_{n-i}\Phi _{i}(w)}$

with

${\displaystyle \Phi _{0}(w)\equiv 1.}$

Thus

${\displaystyle \sum _{n\geq 0}\Phi _{n}(g(z))\zeta ^{-n}=1+\sum _{n\geq 1}\left(z^{n}+\sum _{m\geq 1}c_{nm}z^{-m}\right)\zeta ^{-n},}$

so that for n ≥ 1

${\displaystyle \Phi _{n}(g(z))=z^{n}+\sum _{m\geq 1}c_{nm}z^{-m}.}$

The latter property uniquely determines the Faber polynomial of g.

Let g(z) be a univalent function on |z| > 1 normalized so that

${\displaystyle g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots }$

and let f(z) be a non-constant holomorphic function on C.

If

${\displaystyle f(g(z))=\sum _{-\infty }^{\infty }c_{n}z^{n}}$

is the Laurent expansion on z > 1, then

${\displaystyle \sum _{n>0}n|c_{n}|^{2}\leq \sum _{n>0}n|c_{-n}|^{2}.}$

If

${\displaystyle p(w)=\sum _{n=1}^{N}n^{-1}\lambda _{n}\Phi _{n}(w),}$

then

${\displaystyle p(g(z))=\left(\sum _{n=1}^{N}n^{-1}\lambda _{n}z^{n}\right)+\left(\sum _{m=1}^{\infty }\sum _{n=1}^{N}\lambda _{n}c_{nm}z^{-m}\right).}$

Applying Milin's area theorem,

${\displaystyle \sum _{m=1}^{\infty }m\left|\sum _{n=1}^{N}c_{mn}\lambda _{n}\right|^{2}\leq \sum _{n=1}^{N}{1 \over n}|\lambda _{n}|^{2}.}$

(Equality holds here if and only if the complement of the image of g has Lebesgue measure zero.)

So a fortiori

${\displaystyle \sum _{m=1}^{N}m\left|\sum _{n=1}^{N}c_{mn}\lambda _{n}\right|^{2}\leq \sum _{n=1}^{N}{1 \over n}|\lambda _{n}|^{2}.}$

Hence the symmetric matrix

${\displaystyle a_{mn}={\sqrt {mn}}c_{mn},}$

regarded as an operator on C^N with its standard inner product, satisfies

${\displaystyle \|Ax\|\leq \|x\|.}$

So by the Cauchy–Schwarz inequality

${\displaystyle |(Ax,y)|\leq \|x\|\cdot \|y\|.}$

With

${\displaystyle x_{n}={\frac {\lambda _{n}}{\sqrt {n}}}={\overline {y_{n}}},}$

this gives the Grunsky inequality:

${\displaystyle \left|\sum _{m=1}^{N}\sum _{n=1}^{N}c_{mn}\lambda _{m}\lambda _{n}\right|^{2}\leq \sum _{n=1}^{N}{1 \over n}|\lambda _{n}|^{2},}$

Let g(z) be a holomorphic function on z > 1 with

${\displaystyle g(z)=z+b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots }$

Then g is univalent if and only if the Grunsky coefficients of g satisfy the Grunsky inequalities for all N.

In fact the conditions have already been shown to be necessary. To see sufficiency, note that

${\displaystyle \log {g(z)-g(\zeta ) \over z-\zeta }=-\sum _{m,n\geq 1}c_{mn}z^{-m}\zeta ^{-n}}$

makes sense when |z| and |ζ| are large and hence the coefficients c_mn are defined. If the Grunsky inequalities are satisfied then it is easy to see that the |c_mn| are uniformly bounded and hence the expansion on the left hand side converges for |z| > 1 and |ζ| > 1. Exponentiating both sides, this implies that g is univalent.

Let ${\displaystyle F(z)}$ and ${\displaystyle g(\zeta )}$ be univalent holomorphic functions on |z| < 1 and |ζ| > 1, such that their images are disjoint in C. Suppose that these functions are normalized so that

${\displaystyle g(\zeta )=\zeta +a_{0}+b_{1}\zeta ^{-1}+b_{2}\zeta ^{-2}+\cdots }$

and

${\displaystyle F(z)=af(z)}$

with a ≠ 0 and

${\displaystyle f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots .}$

The Grunsky matrix (c_mn) of this pair of functions is defined for all non-zero m and n by the formulas:

${\displaystyle {\begin{aligned}\log {g(\zeta )-g(\eta ) \over \zeta -\eta }&=-\sum _{m,n\geq 1}c_{mn}\zeta ^{-m}\eta ^{-n}\\\log {g(\zeta )-f(z) \over \zeta }-\log {g(\zeta ) \over \zeta }&=-\sum _{m,n\geq 1}c_{-m,n}z^{m}\zeta ^{-n}\\\log {f(z)-f(w) \over z-w}-\log {f(z) \over z}-\log {f(w) \over w}&=-\sum _{m,n\geq 1}c_{-m,-n}z^{m}w^{n}\end{aligned}}}$

with

${\displaystyle c_{m,-n}=c_{-n,m},\qquad m,n\geq 1,}$

so that (c_mn) is a symmetric matrix.

In 1972 the American mathematician James Hummel extended the Grunsky inequalities to this matrix, proving that for any sequence of complex numbers λ_±1, ..., λ_±N

${\displaystyle \left|\sum _{n,m\neq 0}c_{mn}\lambda _{m}\lambda _{n}\right|\leq \sum _{n\neq 0}{\frac {1}{|n|}}|\lambda _{n}|^{2}.}$

The proof proceeds by computing the area of the image of the complement of the images of |z| < r < 1 under F and |ζ| > R > 1 under g under a suitable Laurent polynomial h(w).

Let ${\displaystyle \phi _{n}}$ and ${\displaystyle \phi _{-n}}$ denote the Faber polynomials of g and ${\displaystyle f(z^{-1})^{-1}}$ and set

${\displaystyle h(w)=\sum _{n\geq 1}{\frac {\lambda _{n}}{n}}\Phi _{n}(w)+\sum _{n\geq 1}{\frac {\lambda _{-n}}{n}}\Phi _{-n}\left({\frac {a}{w}}\right).}$

Then:

${\displaystyle {\begin{aligned}h(F(z))&=\sum _{n\geq 1}{\frac {\lambda _{-n}}{n}}z^{-n}+\alpha +\sum _{n\geq 1}\alpha _{n}z^{n},&&|z|<1,\alpha _{n}=\sum _{m}c_{-n,m}\lambda _{m}\\h(g(\zeta ))&=\sum _{n\geq 1}{\frac {\lambda _{n}}{n}}\zeta ^{n}+\beta +\sum _{n\geq 1}\beta _{n}\zeta ^{-n},&&|\zeta |>1,\beta _{n}=\sum _{m}c_{nm}\lambda _{m}\end{aligned}}}$

The area equals

${\displaystyle \int |h'(z)|^{2}\,dx\,dy={\frac {1}{2i}}\int _{C_{1}}{\overline {h}}(z)h'(z)\,dz-{\frac {1}{2i}}\int _{C_{2}}{\overline {h}}(z)h'(z)\,dz,}$

where C₁ is the image of the circle |ζ| = R under g and C₂ is the image of the circle |z| = r under F.

Hence

${\displaystyle {\frac {1}{\pi }}\iint |h'|^{2}\,dx\,dy=\left[\sum _{n\geq 1}{\frac {1}{n}}|\lambda _{-n}|^{2}-\sum _{n\geq 1}|\alpha _{n}|^{2}r^{2n}\right]+\left[\sum _{n\geq 1}{1 \over n}|\lambda _{n}|^{2}-\sum _{n\geq 1}|\beta _{n}|^{2}R^{-2n}\right].}$

Since the area is positive, the right hand side must also be positive. Letting r increase to 1 and R decrease to 1, it follows that

${\displaystyle \sum _{m\neq 0}|m|\left|\sum _{n\neq 0}c_{mn}\lambda _{n}\right|^{2}\leq \sum _{m\neq 0}{1 \over |m|}|\lambda _{m}|^{2}}$

with equality if and only if the complement of the images has Lebesgue measure zero.

As in the case of a single function g, this implies the required inequality.

The matrix

${\displaystyle a_{mn}={\sqrt {|mn|}}\cdot c_{mn}}$

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

${\displaystyle A^{*}A=I,}$

But if J denotes complex conjugation of a sequence, then

${\displaystyle JAJ=A^{*},\quad JA^{*}J=A}$

since A is symmetric. Hence

${\displaystyle AA^{*}=JA^{*}AJ=I}$

so that A is unitary.

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.[1][2]

In the proof of Schiffer and Charzynski, if

${\displaystyle f(x)=z+a_{2}z^{2}+a_{3}z^{3}+a_{4}z^{4}+\cdots }$

is a normalized univalent function in |z| < 1, then

${\displaystyle g(z)=f(z^{2})^{-1/2}=z+b_{1}z^{-1}+b_{3}z^{-3}+\cdots }$

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 |a₄| in terms of a simple function of a₂ and a free complex parameter. The free parameter can be chosen so that the bound becomes a function of half the modulus of a₂ 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

${\displaystyle \log {g(z)-g(\zeta ) \over z-\zeta }=-\sum _{n\geq 1}a_{n}(\zeta ^{-1})z^{-n}.}$

with a_n(w) holomorphic in |w| < 1.

The Grunsky inequalities, with λ_n = wⁿ imply that

${\displaystyle \sum _{n\geq 1}n|a_{n}(w)|^{2}\leq -\log(1-|w|^{2}).}$

On the other hand, if

${\displaystyle \sum _{m\geq 0}b_{m}t^{m}=\exp \sum _{n\geq 1}a_{n}t^{n}}$

as formal power series, then the first of the Lebedev–Milin inequalities (1965) states that[3][4]

${\displaystyle \sum _{n\geq 0}|b_{n}|^{2}\leq \exp \sum _{n\geq 1}n|a_{n}|^{2}.}$

Equivalently the inequality states that if g(z) is a polynomial with g(0) = 0, then

${\displaystyle {1 \over 2\pi }\int _{0}^{2\pi }|e^{g}|^{2}\,d\theta \leq e^{A},}$

where A is the area of g(D),

To prove the inequality, note that the coefficients are determined by the recursive formula

${\displaystyle b_{n}={1 \over n}\sum _{m=1}^{n}ma_{m}b_{n-m}}$

so that by the Cauchy–Schwarz inequality

${\displaystyle |b_{n}|^{2}\leq {1 \over n}\sum m^{2}|a_{m}|^{2}|b_{n-m}|^{2}.}$

The quantities c_n obtained by imposing equality here:

${\displaystyle c_{n}={1 \over n}\sum m^{2}|a_{m}|^{2}c_{n-m}}$

satisfy ${\displaystyle |b_{n}|^{2}\leq c_{n}}$ and hence, reversing the steps,

${\displaystyle \sum |b_{n}|^{2}\leq \sum c_{n}=\exp \sum _{m\geq 1}m|a_{m}|^{2}.}$

In particular defining b_n(w) by the identity

${\displaystyle \sum b_{n}(\zeta ^{-1})z^{-n}=\exp \sum a_{m}(\zeta ^{-1})z^{-m}={g(z)-g(\zeta ) \over z-\zeta },}$

the following inequality must hold for |w| < 1

${\displaystyle \sum _{n\geq 0}|b_{n}(w)|^{2}\leq (1-|w|^{2})^{-1}.}$

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 Bergman & Schiffer (1951) and Baranov & Hedenmalm (2008).

The Beurling transform is defined on L²(C) as the operation of multiplication by ${\displaystyle z/{\overline {z}}}$ on Fourier transforms. It thus defines a unitary operator. It can also be defined directly as a principal value integral[5]

${\displaystyle (Th)(w)=\lim _{\varepsilon \to 0}-{1 \over \pi }\iint _{|z-w|\geq \varepsilon }{h(z) \over (z-w)^{2}}\,dx\,dy.}$

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 L²(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

${\displaystyle K_{f}(z,w)={\frac {f'(z)f'(w)}{(f(z)-f(w))^{2}}}.}$

It defines a contraction. On the other hand, it can be checked that T_D = 0 by computing directly on powers ${\displaystyle {\overline {z}}^{n}}$ using Stokes theorem to transfer the integral to the boundary.

It follows that the operator with kernel

${\displaystyle {f'(z)f'(w) \over (f(z)-f(w))^{2}}-{1 \over (z-w)^{2}}={\partial ^{2} \over \partial z\partial w}\log {f(z)-f(w) \over z-w}=-\sum _{m,n\geq 1}mnc_{mn}z^{m-1}w^{n-1}}$

acts as a contraction on the conjugate of the Bergman space of D. Hence, if

${\displaystyle p(z)=\lambda _{1}+\lambda _{2}{\overline {z}}+\lambda _{3}{\overline {z}}^{2}+\cdots +\lambda _{N}{\overline {z}}^{N-1},}$

then

${\displaystyle \sum _{m=1}^{N}\left|\sum _{n=1}^{N}c_{mn}\lambda _{n}\right|^{2}=\|(T_{f}-T_{z})p\|^{2}=\|T_{f}p\|^{2}\leq \|p\|^{2}=\sum _{n=1}^{N}{1 \over n}|\lambda _{n}|^{2}.}$

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 = A²(Ω). It is given by the formula

${\displaystyle (T_{\Omega }u)(z)=\lim _{\varepsilon \to 0}{1 \over \pi }\iint _{|z-w|\geq \varepsilon }{{\overline {u(z)}} \over (z-w)^{2}}\,\,dx\,dy}$

for u in the Hilbert space H= A²(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 T_D = 0 shows that it is given by restriction of the kernel

${\displaystyle {\frac {f'(z)f'(w)}{(f(z)-f(w))^{2}}}-{\frac {1}{(z-w)^{2}}},}$

and is therefore a Hilbert–Schmidt operator.

The antilinear operator T = T_Ω satisfies the self-adjointness relation

${\displaystyle (Tu,v)=(Tv,u)}$

for u, v in H.

Thus A = T² is a compact self-adjont linear operator on H with

${\displaystyle (Au,u)=(Tu,Tu)=\|Tu\|^{2}\geq 0,}$

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:

${\displaystyle Au_{n}=\mu _{n}u_{n},}$

where μ_n is non-negative by the positivity of A. Hence

${\displaystyle \mu _{n}=\lambda _{n}^{2}}$

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:

${\displaystyle Tu_{n}=\lambda _{n}u_{n}.}$

(Note that ${\displaystyle T(iu_{n})=-\lambda _{n}iu_{n}}$ by antilinearity of T.)

The non-zero λ_n (or sometimes their reciprocals) are called the Fredholm eigenvalues of Ω:

${\displaystyle 0\leq \lambda _{n}\leq \|T\|\leq 1.}$

If Ω is a bounded domain that is not a disk, Ahlfors showed that

${\displaystyle \|T_{\Omega }\|<1.}$

The Fredholm determinant for the domain Ω is defined by[6][7]

${\displaystyle \Delta _{\Omega }=\det(I-T_{\Omega }^{2})=\prod (1-\lambda _{n}^{2}).}$

Note that this makes sense because A = T² is a trace class operator.

Schiffer & Hawley (1962) showed that if ${\displaystyle 0\in \Omega }$ and f fixes 0, then[8][9]

${\displaystyle \Delta _{\Omega }=-{\frac {1}{12\pi }}\left[\|\partial _{z}\log f'\|_{D}^{2}+\|\partial _{z}\log g'\|_{D^{c}}^{2}-2\left\|\partial _{z}\log {\frac {f(z)}{z}}\right\|_{D}^{2}-2\left\|\partial _{z}\log {\frac {g(z)}{z}}\right\|_{D^{c}}^{2}\right].}$

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).

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 S¹ and C.

• Ahlfors, Lars V. (1952), "Remarks on the Neumann-Poincaré integral equation", Pacific J. Math., 2 (3): 271–280, doi:10.2140/pjm.1952.2.271
• Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
• Ahlfors, Lars V. (2010), Conformal invariants. Topics in geometric function theory. Reprint of the 1973 original. With a foreword by Peter Duren, F. W. Gehring and Brad Osgood, AMS Chelsea Publishing, ISBN 978-0-8218-5270-5
• Astala, Kari; Iwaniec, Tadeusz; Martin, Gaven (2009), Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton mathematical series, vol. 48, Princeton University Press, ISBN 978-0-691-13777-3
• Baranov, A.; Hedenmalm, H. (2008), "Boundary properties of Green functions in the plane", Duke Math. J., 145: 1–24, arXiv:math/0608493, doi:10.1215/00127094-2008-044, S2CID 53692019
• Bell, S. R. (1992), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics, CRC Press, ISBN 978-0-8493-8270-3
• Bell, S. R. (2016), The Cauchy transform, potential theory, and conformal mapping, Studies in Advanced Mathematics (2nd ed.), CRC Press, ISBN 9781498727211
• Bergman, S.; Schiffer, M. (1951), "Kernel functions and conformal mapping", Compositio Mathematica, 8: 205–249
• Duren, P. L. (1983), Univalent functions, Grundlehren der Mathematischen Wissenschaften, vol. 259, Springer-Verlag, ISBN 978-0-387-90795-6
• Gakhov, F. D. (1990), Boundary value problems. Reprint of the 1966 translation, Dover Publications, ISBN 978-0-486-66275-6
• Garnett, J. B. (2007), Bounded analytic functions, Graduate Texts in Mathematics, vol. 236, Springer, ISBN 978-0-387-33621-3
• Goluzin, G. M. (1969), Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society
• Gong, Sheng (1999), The Bieberbach conjecture, AMS/IP Studies in Advanced Mathematics, vol. 12, American Mathematical Society, ISBN 978-0-8218-0655-5
• Grinshpan, A. Z. (1999), "The Bieberbach conjecture and Milin's functionals", The American Mathematical Monthly, 106 (3): 203–214, doi:10.2307/2589676, JSTOR 2589676, MR 1682341
• Grinshpan, Arcadii Z. (2002), "Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains", in Kuhnau, Reiner (ed.), Geometric Function Theory, Handbook of Complex Analysis, vol. 1, Amsterdam: North-Holland, pp. 273–332, ISBN 978-0-444-82845-3, MR 1966197, Zbl 1083.30017.
• Grunsky, Helmut (1939), "Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen", Mathematische Zeitschrift, 45 (1): 29–61, doi:10.1007/BF01580272, ISSN 0025-5874, S2CID 123606166
• Grunsky, Helmut (1978), Lectures on theory of functions in multiply connected domains, Studia Mathematica, vol. 4, Vandenhoeck & Ruprecht, ISBN 978-3-525-40142-2
• Hayman, W. K. (1994), "De Branges' Theorem", Multivalent functions, Cambridge Tracts in Mathematics, vol. 110 (2nd ed.), Cambridge University Press, ISBN 0521460263
• Khavinson, D.; Putinar, M.; Shapiro, H. S. (2007), "Poincaré's variational problem in potential theory", Arch. Ration. Mech. Anal., 185 (1): 143–184, Bibcode:2007ArRMA.185..143K, CiteSeerX 10.1.1.569.7145, doi:10.1007/s00205-006-0045-1, S2CID 855706
• Koepf, W. (2007), "Bieberbach's conjecture, the de Branges and Weinstein functions and the Askey-Gasper inequality" (PDF), The Ramanujan Journal, 13 (1–3): 103–129, doi:10.1007/s11139-006-0244-2, S2CID 16263023
• Milin, I. M. (1977), Univalent functions and orthonormal systems, Translations of Mathematical Monographs, vol. 49, American Mathematical Society
• Neretin, Y. A. (1996), Categories of symmetries and infinite-dimensional groups, London Mathematical Society Monographs, vol. 16, Oxford University Press, ISBN 978-0-19-851186-1
• Nikolski, N. K. (2002), Operators, functions, and systems: an easy reading, Vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, ISBN 978-0-8218-1083-5
• Pommerenke, C. (1975), Univalent functions, with a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, vol. 15, Vandenhoeck & Ruprecht
• Schiffer, M. (1948), "Faber polynomials in the theory of univalent functions", Bull. Amer. Math. Soc., 54 (6): 503–517, doi:10.1090/S0002-9904-1948-09027-9
• Schiffer, M. (1957), "The Fredholm eigenvalues of plane domains", Pacific J. Math., 7 (2): 1187–1225, doi:10.2140/pjm.1957.7.1187
• Schiffer, M. (1959), "Fredholm eigenvalues of multiply connected domains", Pacific J. Math., 9: 211–269, doi:10.2140/pjm.1959.9.211
• Schiffer, M.; Hawley, N. S. (1962), "Connections and conformal mapping", Acta Math., 107 (3–4): 175–274, doi:10.1007/bf02545790
• Schiffer, M. (1981), "Fredholm eigenvalues and Grunsky matrices", Ann. Polon. Math., 39: 149–164, doi:10.4064/ap-39-1-149-164
• Schur, I. (1945), "On Faber polynomials", Amer. J. Math., 67 (1): 33–41, doi:10.2307/2371913, JSTOR 2371913
• Shapiro, H. S. (1992), The Schwarz function and its generalization to higher dimensions, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 9, Wiley-Interscience, ISBN 978-0-471-57127-8
• Takhtajan, Leon A.; Teo, Lee-Peng (2006), "Weil–Petersson metric on the universal Teichmüller space", Mem. Amer. Math. Soc., 183
• Widom, H. (1988), "On an inequality of Osgood, Phillips and Sarnak", Proc. Amer. Math. Soc., 102 (3): 773–774, doi:10.1090/s0002-9939-1988-0929019-3