Distortion theorem for bounded univalent functions |
Pavlov N.A. |
2015, issue 2, P. 238-246 |
Abstract |
Distortion theorem for bounded univalent functions in the unit disk is proven. Estimate includes angular derivatives in two boundary points and Schwarzian derivative in interior point of the unit disk. |
Keywords: univalent functions, angular derivative, Schwarzian derivative, capacity of condenser |
Download the article (PDF-file) |
References |
[1] Ch. Pommerenke, Boundary behaviour of conformal maps, Springer, Berlin, 1992. [2] C.C. Cowen, Ch. Pommerenke, "Inequalities for the angular derivative of an analytic function in the unit disk", J. London Math. Soc., 2:26 (1982), 271-289. [3] A. Frolova, M. Levenshtein, D. Shoikhet, A. Vasil'ev, "Boundary distortion estimates for holomorphic maps", Complex Analysis and Operator Theory, 8:5 (2014), 1129-1149. [4] Ch. Pommerenke, A. Vasil'ev, "Angular derivatives of bounded univalent functions and extremal partitions of the unit disk", Pacific. J. Math., 206:2 (2002), 425-450. [5] Z. Nehari, "Some inequalities in the theory of functions", Trans. Amer. Math. Soc., 75:2 (1953), 256-286. [6] V.N. Dubinin, Emkosti kondensatorov i simmetrizacija v geometricheskoj teorii funkcij kompleksnogo peremennogo, Dal'nauka, Vladivostok, 2009. [7] I.I. Privalov, Vvedenie v teoriju funkcij kompleksnogo peremennogo, Nauka, Glavnaja redakcija fiziko-matematicheskoj literatury, M., 1984. |