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

We generalized some criteria of boundedness of L-index in joint variables for analytic functions in a complete multiple circular domain, where L(z)=(l₁(z₁,z₂,…,zₙ), l₂(z₁,z₂,…,zₙ), …, lₙ(z₁,z₂,…,zₙ)), lⱼ G→ R₊ is a continuous function, G is the n-dimensional complete multiple circular domain in Cⁿ, i.e. for every point (z₁,,zₙ) from this domain G and for each rⱼ∈[0,1], θ∈[0,2π], j∈\1,2,…,n\, the point-wise product (r₁z₁,,rₙzₙ) belongs to the same domain G and the component-wise rotation (z₁eiθ₁,…, zₙeiθₙ) falls into this domain 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 G.Replacing universal quantifier by existential quantifier for radii of bidisc, we also proved sufficient conditions of boundedness of L-index in joint variables for analytic functions which are weaker than necessary conditions.