Dominating polynomial in power series expansion for analytic functions in a complete Reinhardt domain

Abstract

We generalized some criteria of boundedness of $\mathbf{L}$-index in joint variables for analytic functions in a complete multiple circular domain,  where $\mathbf{L}(z)=(l_1(z_1,z_2,\ldots,z_n),$ $l_{2}(z_1,z_2,\ldots,z_n),$ $\ldots,$ $l_{n}(z_1,z_2,\ldots,z_n)),$  $l_j \colon \mathbb{G}\to \mathbb{R}_+$ is a  continuous function,  $\mathbb{G}$ is the  $n$-dimensional complete multiple circular domain in $\mathbb{C}^n,$ i.e.  for every point $(z_1,\dots,z_n)$ from this domain $\mathbb{G}$ and for each $r_j\in[0,1],$ $\theta\in[0,2\pi],$ $j\in\{1,2,\ldots,n\},$ the point-wise product $(r_1z_1,\dots,r_nz_n)$ belongs to the same domain $\mathbb{G}$ and the component-wise rotation $(z_1e^{i\theta_1},\ldots, z_ne^{i\theta_n})$ falls into this domain $\mathbb{G}.$ The propositions describe a behavior of multiple power series expansion on a skeleton of a polydisc. There are presented estimation of power series expansion modulus by a dominating homogeneous polynomial with the degree that does not exceed some number depending only from radii of polydisc. Changing the center of the polydisc, we cover the whole domain $\mathbb{G}.$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.