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

  • A.I. Bandura Ivano-Frankivsk National Technical University of Oil and Gas, 15 Karpatska str., 76019, Ivano-Frankivsk, Ukraine
  • N.V. Petrechko Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
Keywords: analytic function, bidisc, bounded $\mathbf{L}$-index in joint variables, maximum modulus, partial derivative, dominating polynomial, power series
Published online: 2017-06-08


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.

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