Abstract
In the present paper we generalize Carathéodory’s inequality for functions holomorphic in Cartan domains in Cn. In particular, in the case of functions holomorphic in the unit disk in C, this generalization of Carathéodory’s inequality implies the classical inequalities of Carahtéodory and Landau. As an application, new results on multidimensional analogues of Bohr’s theorem on power series are obtained. Furthermore, the estimate from below of Bohr radius is improved for the domain \(D = \{ z \in C^2 :\left| {z_1 } \right| + \left| {z_2 } \right| < 1\}\).
During the preparation of this work the author was supported by the Israel-Slovenia grant. Part of the work was completed in January-February 2001 during the author’s stay at the Institute of Mathematics, Physics and Mechanincs, Lujbljiana, Slovenia.
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© 2005 Birkhäuser Verlag Basel/Switzerland
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Aizenberg, L. (2005). Generalization of Carathéodory’s Inequality and the Bohr Radius for Multidimensional Power Series. In: Eiderman, V.Y., Samokhin, M.V. (eds) Selected Topics in Complex Analysis. Operator Theory: Advances and Applications, vol 158. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7340-7_6
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DOI: https://doi.org/10.1007/3-7643-7340-7_6
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