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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Grunsky inequalities ： ウィキペディア英語版 Grunsky matrix In mathematics, the Grunsky matrices, or Grunsky operators, are matrices introduced by 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 function :$g(z)\; =\; f(z^)^$ is a non-vanishing univalent function on |''z''| > 1 having a simple pole at ∞ with residue 1: :$g(z)\; =\; z\; +\; b\_0\; +\; b\_1z^\; +\; b\_2\; z^\; +\; \backslash cdots$ 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 :$\backslash log\backslash frac\; =\; -\backslash sum\_c\_z^\backslash zeta^$ It is a symmetric matrix. Its entries are called the Grunsky coefficients of ''g''. Note that :$\backslash log)\backslash over\; z^-\backslash zeta^\}\; =\; \backslash log\; -\backslash log\; -\backslash log\; ,$ so that that the coefficients can be expressed directly in terms of ''f''. Indeed if :$\backslash log\; =\; -\backslash sum\_\; d\_\; z^n\; \backslash zeta^n,$ then for ''m'', ''n'' > 0 :$\backslash displaystyle\}$ and ''d''0''n'' = ''d''''n''0 is given by :$\backslash log\; =\; \backslash sum\_\; d\_\; z^n$ with :$\backslash displaystyle$ ==Grunsky inequalities== If ''f'' is a holomorphic function on the unit disk with Grunsky matrix (''c''''nm''), the Grunsky inequalities state that :$\backslash left|\backslash sum\_\; c\_\backslash lambda\_m\; \backslash lambda\_n\; \backslash right|\backslash le\; \backslash sum\_\; |\backslash lambda\_n|^2/n$ for any finite sequence of complex numbers λ1, ..., λ''N''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Grunsky matrix」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.