The regularity theorem is a result stating that functions which have extremal growth or decrease in the given class display a regular behaviour. Such theorems for linearly invariant families of analytic functions are well known. We prove regularity theorems for some classes of harmonic functions. Many presented statements are new even in the analytic case.

In this article, we present univalence criteria for polyharmonic and polyanalytic functions. Our approach yields new a criterion for a polyharmonic functions to be fully α-accessible. Several examples are presented to illustrate the use of these criteria.2000 Mathematics Subject Classification. Primary: 31A05, 31A30, 31C05, 30C45, 52A30; Secondary: 30C20.

Abstract.In 1954 M. Heins proved that, for every analytic set

The article is devoted to the class A α,β ρ of all (α, β)accessible with respect to the origin domains D, α, β ∈ [0, 1), possessing the property ρ = min p∈∂D |p|, where ρ ∈ (0, +∞) is a fixed number. We find the maximal set of points a such that all domains D ∈ A α,β ρ are (γ, δ)-accessible with respect to a, γ ∈ [0; α], δ ∈ [0; β]. This set is proved to be the closed disc of center 0 and radius ρ sin φπ 2 , where φ = min {α − γ, β − δ} .

The classical theorem of growth regularity in the class S of analytic and univalent in the unit disc ∆ functions f describes the growth character of different functionals of f ∈ S and z ∈ ∆ as z tends to ∂∆. Earlier the authors proved the theorems of growth and decrease regularity for harmonic and sensepreserving in ∆ functions which generalized the classical result for the class S. In the presented paper we establish new properties of harmonic sense-preserving functions, connected with the regularity theorems. The effects both common for analytic and harmonic case and specific for harmonic functions are displayed.