Properties of power series of analytic in a bidisc functions of bounded $\mathbf{L}$-index in joint variables

A. I. Bandura, N. V. Petrechko


We generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic in a bidisc functions, where $\mathbf{L}(z)=(l_1(z_1,z_2),$ $l_{2}(z_1,z_2)),$ $l_j:\mathbb{D}^2\to \mathbb{R}_+$ is a continuous function, $j\in\{1,2\},$ $\mathbb{D}^2$ is a bidisc $\{(z_1,z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1\}.$ The propositions describe a behaviour of power series expansion on a skeleton of a bidisc. We estimated power series expansion by a dominating homogeneous polynomial with the degree that does not exceed some number depending only from radii of bidisc. Replacing universal quantifier by existential quantifier for radii of bidisc, we also proved sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables for analytic functions which are weaker than necessary conditions.


analytic function, bidisc, bounded $\mathbf{L}$-index in joint variables, maximum modulus, partial derivative, dominating polynomial, power series

Full Text: Article References
3 :: 9


  • There are currently no refbacks.

Creative Commons License
The journal is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported.