Analytic Functions in a Complete Reinhardt Domain Having Bounded L-Index in Joint Variables

Abstract

The manuscript is an initiative to construct a full and exhaustive theory of analytical multivariate functions in any complete Reinhardt domain by introducing the concept of L-index in joint variables for these functions for a given continuous, non-negative, non-vanishing, vector-valued mapping L defined in an interior of the domain with some behavior restrictions. The complete Reinhardt domain is an example of a domain having a circular symmetry in each complex dimension. Our results are based on the results obtained for such classes of holomorphic functions: entire multivariate functions, as well as functions which are analytical in the unit ball, in the unit polydisc, and in the Cartesian product of the complex plane and the unit disc. For a full exhaustion of the domain, polydiscs with some radii and centers are used. Estimates of the maximum modulus for partial derivatives of the functions belonging to the class are presented. The maximum is evaluated at the skeleton of some polydiscs with any center and with some radii depending on the center and the function L and, at most, it equals a some constant multiplied by the partial derivative modulus at the center of the polydisc. Other obtained statements are similar to the described one.

Uniform estimates for local properties of analytic functions in a complete Reinahrdt domain

“…A multivariate holomorphic function H ∈ A(G) is called a function with bounded (finite) L-index (in joint variables) (see [1]) if, for some non-negative integer n 0 , the following inequality holds for every order J of partial derivatives in the whole domain G:…”