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- Grunsky_matrix abstract "In mathematics, the Grunsky matrices, or Grunsky operators, are matrices introduced by Grunsky (1939) in complex analysis and geometric function theory. They 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. Historically the inequalities 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. The Grunsky operators and their Fredholm determinants are 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.If f(z) is a holomorphic univalent function on the unit disk, normalized so that f(0) = 0 and f'(0) = 1, the functionis 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 correspondencebetween these two classes of function.The Grunsky matrix (cnm) of g is defined by the equationIt is a symmetric matrix. Its entries are called the Grunsky coefficients of g.Note thatso that that the coefficients can be expressed directly in terms of f. Indeed ifthen for m, n > 0and d0n = dn0 is given bywith".
- Grunsky_matrix wikiPageExternalLink BF01580272.
- Grunsky_matrix wikiPageExternalLink Koepf_Bexbach2003.pdf.
- Grunsky_matrix wikiPageID "33994467".
- Grunsky_matrix wikiPageRevisionID "599328793".
- Grunsky_matrix hasPhotoCollection Grunsky_matrix.
- Grunsky_matrix subject Category:Complex_analysis.
- Grunsky_matrix subject Category:Moduli_theory.
- Grunsky_matrix subject Category:Operator_theory.
- Grunsky_matrix comment "In mathematics, the Grunsky matrices, or Grunsky operators, are matrices introduced by Grunsky (1939) in complex analysis and geometric function theory. They 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.".
- Grunsky_matrix label "Grunsky matrix".
- Grunsky_matrix sameAs m.0hr2lqj.
- Grunsky_matrix sameAs Q5612135.
- Grunsky_matrix sameAs Q5612135.
- Grunsky_matrix wasDerivedFrom Grunsky_matrix?oldid=599328793.
- Grunsky_matrix isPrimaryTopicOf Grunsky_matrix.