Wiman-Type Inequality in Multiple-Circular Domains: Lévy’s Phenomenon and Exceptional Sets

Kuryliak, A. O.; Скасків, О. Б.

doi:10.1007/s11253-022-02098-y

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“…An increasing number of papers on various types of Reinhardt domains [6], on the Schwarz lemma [7], on the rigidity theorem [8], on Bohr radii [9,10], on Bergman kernels [11], and on the bounds of all the coefficients of homogeneous expansions [12] for the domain show the importance of this topic. A few recent papers [13,14] initiated an intensive study of functions that are analytical in a complete Reinhardt domain by methods of the Wiman-Valiron theory. We will take most of our notations from those papers: R n and C n have the typical meanings in complex analysis, i.e, they are the real and complex n-dimensional vector spaces, respectively.…”

Section: Introduction Main Notations and Definitionsmentioning

confidence: 99%

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

Bandura,

Salo,

Skaskiv

2024

Symmetry

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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.

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Section: Introduction Main Notations and Definitionsmentioning

confidence: 99%

“…Recent publications [3,10,13,14] show the growth of adaptations of one-dimensional complex analytic methods to such a domain as the Reinhardt domain in multidimensional complex analysis. Its increasing value is justified by its properties.…”

Section: Introduction Main Notations and Definitionsmentioning

confidence: 99%