Properties of power series of analytic in a bidisc functions of bounded $\mathbf{L}$-index in joint variables
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 1
Abstract
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 $\{(z_1,z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1\}.$ We obtained propositions, which describe a behaviour of power series expansion on a skeleton of a bidisc. The power series expansion is estimated by a dominating homogeneous polynomial with a degree that does not exceed some number, depending only from radii of a bidisc. Replacing universal quantifier by existential quantifier for radii of a bidisc, we also proved sufficient conditions of boundedness of $\mathbf{L}$-index in joint variables for analytic functions, which are weaker than necessary conditions.
Authors and Affiliations
A. Bandura, N. Petrechko
Keywords
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- DOI 10.15330/cmp.9.1.6-12
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