Ekaterina Ganenkova

• PhD
• Docent at Petrozavodsk State University

The question raised in this article goes back to the problem posed by the famous chemist D. I. Mendeleev in 1887 (solved by A. A. Markov in 1889). In the next 100 years, the Mendeleev problem was repeatedly modificated and solved. Its essence is in the description of conditions under which the inequality ∣f(z)∣ ≤ ∣F(z)∣ for polynomials f and F and...

Differential inequalities for polynomials generalizing the well-known Smirnov, Rahman, Schmeisser, and Bernstein inequalities are obtained.

In 1949 Marden in his book introduced and studied the differential operator B[f](z)=λ 0 f(z)+λ 1 [Formula presented]f ′ (z)+λ 2 ([Formula presented]) ² [Formula presented], defined on polynomials of degree at most n. Here λ 0 ,λ 1 , and λ 2 are constants such that the polynomial u(z)=λ 0 +C n¹ λ 1 z+C n² λ 2 z ² has all roots in the half-plane Rez≤...

In this article, we present univalence criteria for polyharmonic and polyanalytic functions. Our approach yields a new criterion for a polyharmonic functions to be fully -accessible. Several examples are presented to illustrate the use of these criteria.

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 $\alpha$--accessible. Several examples are presented to illustrate the use of these criteria.

Asymptotic sets of functions in a polydisk domain of arbitrary connectivity are studied. We construct an example of such function, having preassigned asymptotic set. This result generalizes well-known examples, obtained by M. Heins and W. Gross for entire functions. Moreover, it is found out that not all results on asymptotic sets of functions in C...

The classical theorem of growth regularity in the class S of analytic and univalent in the unit disc ∆ functions ƒ describes the growth character of different functionals of ƒ Є S and z Є ∆ as z tends to δ∆. Earlier the authors proved the theorems of growth and decrease regularity for harmonic and sense-preserving in ∆ functions which generalized t...

The class A ρα of domains D ⊂ ℝn which are α-accessible with respect to the origin, α ∈ [0, 1], and have the property \(\rho = \mathop {\min }\limits_{\rho \in \partial D} ||p||\), where ρ ∈ (0, +∞) is a fixed number, is considered. Such domains satisfy the so-called cone condition, i.e., are conically accessible from the interior. The maximal set...

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 analyt...

The article is devoted to the class A α,βρ of all (α, β)- accessible with respect to the origin domains D, α, β ∈ [0, 1),possessing the property ρ = minp∈∂D |p|, where ρ ∈ (0, +∞) is a fixed number. We find the maximal set of points a such that all domains D ∈ A α,βρ are (γ, δ)-accessible with respect to a, γ ∈ [0; α], δ ∈ [0; β]. This set is prove...

In 1954 M. Heins proved that, for every analytic set A containing the infinity, there exists an entire function whose set of asymptotic values at the infinity equals A. We obtain analogs of this result for functions analytic in planar domains of arbitrary connectivity.

It is known that an arbitrary function in the open unit disk can have at most countable set of ambiguous points. Point ζ on the unit circle is an ambiguous point of a function if there exist two Jordan arcs, lying in the unit ball, except the endpoint ζ, such that cluster sets of function along these arcs are disjoint. We investigate whether it is...

In [Math. Ann. 79, 201–208 (1918; JFM 46.0512.04)] W. Gross constructed an example of an entire function of infinite order whose set of asymptotic values is equal to the extended complex plane. We obtain an analog of Gross’ result for functions analytic in planar domains of arbitrary conectivity with an isolated boundary fragment.

In this paper we prove an analog of the Bagemihl theorem for functions defined in a polydisk. We apply the obtained result for studying properties of functions of linearly invariant families. Keywords and phrasescluster set–ambiguous point–linearly invariant family