Boundary characteristics of meromorphic functions with summable spherical derivation and annular functions. Consideration

In this paper we formulate classical theorems Plesner and Meyer on the boundary behavior of meromorphic functions and their refinement and strengthening - Gavrilov's and Kanatnikov's theorems. An application of these theorems to classes of meromorphic functions with integrable spherical derivative and annular holomorphic functions is presented. Collingwood's theorem on boundary singularities of the Tsuji function as well as Kanatnikov's theorems are formulated. Kanatnikov's theorems strengthen and generalize Collingwood's theorem to broader classes of meromorphic functions with summable spherical derivatives. Special attention is paid to the boundary properties of annular holomorphic functions. The behavior of annular holomorphic functions on the boundary of the unit circle is considered. It is shown that Gavrilov's P-sequences play an important role in the study of the boundary properties of holomorphic and meromorphic functions.

Generalized Derivatives of an Arbitrary Order and the Boundary Properties of Differentiated Poisson Integrals for the Half-Space

The notions of a generalized differential and a generalized spherical derivative of an arbitrary order are introduced for a function of several variables and Fatou type theorems are proved on the boundary properties of partial derivatives of an arbitrary order of the Poisson integral for the half-space, when the integral density has a generalized differential or a generalized spherical derivative.