|
|
Pure Mathematics 2023
关于一类单叶函数Schwarz导数的注记
|
Abstract:
利用Hilbert空间上的一个有界算子和单叶函数的性质，讨论一类单叶函数的Schwarz导数，并引入一类Grunsky系数。得到有界算子的内积与一类单叶函数Schwarz导数的关系，以及其Schwarz导数在复Hilbert空间下的范数与Grunsky系数的关系。
By using a bounded operator on a Hilbert space and the properties of univalent functions, the Schwarz derivative of a class of univalent functions is discussed and a class of Grunsky coefficients is introduced. The relation between the inner product of a bounded operator and the Schwarz de-rivative of a class of univalent functions is obtained, as well as the relation between the norm of its Schwarz derivative in complex Hilbert space and the Grunsky coefficients.
| [1] | Koebe, P. (1907) Ueber die Uniformisierung beliebiger analytischer Kurven. Nachrichten von der Gesellschaft der Wissenschaften zuG？ttingen. Mathematisch-Physikalische Klasse, 1907, 191-210. http://eudml.org/doc/58678 |
| [2] | Gronwall, T.H. (1914/1915) Some Remarks on Conformal Representation. Annals of Mathematics, 16, 72-76.
https://doi.org/10.2307/1968044 |
| [3] | Grunsky, H. (1939) Koeffizientenbedingungen für schlicht abbildende meromorphe Funktionen. Mathematische Zeitschrift, 45, 29-61. https://doi.org/10.1007/BF01580272 |
| [4] | Sergeev, A. (2014) Lectures on Universal Teichmüller Space. Zürich: European Mathematical Society.
https://doi.org/10.4171/141 |
| [5] | Todorov, P.G. (1979) New Explicit Formulas for the Coefficients of P-Symmetric Functions. Proceedings of the American Mathematical Society, 77, 81-86. https://doi.org/10.1090/S0002-9939-1979-0539635-3 |
| [6] | Hummel, J.A. (1964) The Grunsky Coefficients of a Schlicht Functions. Proceedings of the American Mathematical Society, 15, 142-150.
https://doi.org/10.1090/S0002-9939-1964-0158060-X |
| [7] | Harmelin, R. (1982) Bergman Kernel Function and Univalence Criteria. Journal d’Analyse Mathématique, 41, 249-258.
https://doi.org/10.1007/BF02803404 |
| [8] | Aharonov, D. (1969) A Necessary and Sufficient Condition for Univalence of a Meromorphic Function. Duke Mathematical Journal, 36, 599-604. https://doi.org/10.1215/S0012-7094-69-03671-0 |
| [9] | Shen, Y. (2007) On Grunsky Operator. Science in China Series A: Mathematics, 50, 1805-1817.
https://doi.org/10.1007/s11425-007-0141-1 |
