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

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 $\{(z_1,z_2)\in\mathbb{C}^2: |z_1|<1,|z_2|<1\}.$ 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.