Lev Aizenberg

A Bohr phenomenon for elliptic equations

In Bohr proved that there is an r such that if a power series converges in the unit disk and its sum has modulus less than then for jzj r the sum of absolute values of its terms is again less than Recently analogous results were obtained for functions of several variables The aim of this paper is to comprehend the theorem of Bohr in the context of solutions to second order elliptic equations meeting the maximum principle

Generalization of Carathéodory’s Inequality and the Bohr Radius for Multidimensional Power Series

In the present paper we generalize Caratheodory’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 Caratheodory’s inequality implies the classical inequalities of Carahteodory 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 \(D = \{ z \in C^2 :\left| {z_1 } \right| + \left| {z_2 } \right| < 1\}\).

Boundary Behavior of Semigroups of Holomorphic Mappings on the Unit Ball in C n

We study the asymptotic behavior of semigroups generated by holomorphic mappings by using an infinitesimal version of the boundary Schwarz-Wolff Lemma. In particular, the best rate of exponential convergence is obtained. In addition, we establish a geometrical version of the implicit function theorem.

On Small Complete Sets of Functions

Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T. Carleman and A. F. Leontiev is proven for the space of holomorphic functions defined on a suitable open strip T� ⊂ C2. The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in C n .