# Volume 71, № 5, 2019

### Bernstein – Walsh-type polynomial inequalities in domains bounded by piecewise asymptotically conformal curve with nonzero inner angles in the Bergman space

Abdullayev G. A., Abdullayev F. G., Şimşek D.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 583-595

UDC 517.5

We continue the investigation of the order of growth of the modulus of an arbitrary algebraic polynomial in the Bergman
weight space, where the contour and weight functions have certain singularities. In particular, we deduce a Bernstein–
Walsh-type pointwise estimate for algebraic polynomials in unbounded domains with a piecewise asymptotically conformal
curve with nonzero inner angles in the Bergman weight space.

### New criterion for the analyticity of a function: representation via the metric tensors of the surfaces $Z = u, Z = v$

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 596-609

UDC 514.75:517.53

We establish a new criterion for the analyticity of a function $w=u+iv$ or
$\overline{w}=u-iv,$ $u(x, y),$ $v(x, y) \in C^1(G)$ in the domain $G.$
It is expressed via the metric tensors of the surfaces $Z=u$ and $Z=v\colon g_{11}-a_{22}=0,$ $g_{12}+a_{12} =0,$ $g_{22}-a_{11}=0.$
We also discover some other equivalents of the analytic function and establish the invariance of the obtained relations under conformal transformations.
The generalized version of the new criterion is also proposed.

### Existence results for doubly nonlinear parabolic equations with two lower order terms and $L^1$-data

Benkirane A., El Hadfi Y., El Moumni M.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 610-630

UDC 517.9

We investigate the existence of a renormalized solution for a class of nonlinear parabolic equations with two lower order terms and $L^1$-data.

### Dividend payments in a perturbed compound Poisson model with stochastic investment and debit interest

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 631-644

UDC 519.21

We consider a compound Poisson insurance risk model perturbed by diffusion with stochastic return on investment and debit interest. If the initial surplus is nonnegative, then the insurance company can invest its surplus in a risky asset and risk-free asset based on a fixed proportion. Otherwise, the insurance company can get the business loan when the surplus is negative. The integrodifferential equations for the moment generating function of the cumulative dividends value are obtained under the barrier and threshold dividend strategies, respectively. The closed-form of the expected dividend value is obtained when the claim amount is exponentially distributed.

### Boundary-value problem with impulsive action for a parabolic equation with degeneration

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 645-655

UDC 517.984.54

For a second-order parabolic equation, we consider a problem with oblique derivative and impulsive action. The coefficients of the equation and the boundary condition have power singularities of any order in the time and space variables on some set of points.

We establish conditions for the existence and uniqueness of the solution of the problem in Hölder spaces with power weight.

### Form and properties of the canonical Weierstrass product of an entire function with real values on $\mathbb{R}$

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 656-662

UDC 517.2

We determine the form and properties of the Weierstrass canonical product of an entire function with real values on $\mathbb{R}$.
The proof of the theorem on the order of $R$-integral is specified

[${\it Samoilenko\, A. M.}$ Order and canonical product of the Weierstrass $R$-integral // Ukr. Mat. Zh. — 2019. — ${\bf 71}$, № 4. — P.564-570].

### On the local behavior of Sobolev classes on two-dimensional Riemannian manifolds

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 663-676

UDC 517.9

We study open discrete maps of two-dimensional Riemannian manifolds from the Sobolev class. For these mappings,
we obtain the lower estimates of distortions of the moduli of the families of curves. As a consequence, we establish the
equicontinuity of Sobolev classes at interior points of the domain.

### Investigation of systems of differential equations with delays and constraints imposed on the delays and derivatives of the solutions

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 677-691

UDC 517.929; 517.958:531–133

We establish conditions for the existence and uniqueness of the solutions of nonlinear systems of differential equations with delays and restrictions imposed on the delays and derivatives of the solutions.

### Model of stationary diffusion with absorption in domains with fine-grained random boundaries

Khilkova L. O., Khruslov E. Ya.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 692-705

UDC 517.95, 519.21

We consider a boundary-value problem for the equation of stationary diffusion in a porous medium filled with small ball inclusions with absorbing surfaces. Absorption is described by a Robin’s nonlinear boundary condition. The locations and radii of the inclusions are randomly distributed and described by a set of finite-dimensional distribution functions. We study the asymptotic behavior of solutions to the problem when the number of balls increases and their radii decrease. We derive a homogenized equation for the main term of the asymptotics, and determine sufficient conditions for the distribution functions under which the solutions converge to the solutions of the homogenized problem in probability.

### Structural stability of matrix pencils and of matrix pairs under contragredient equivalence

García-Planas M. I., Klymchuk T.

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 706-709

UDC 512.64

A complex matrix pencil $A-\lambda B$ is called structurally stable if there exists its neighborhood in which all pencils are strictly equivalent to this pencil. We describe all complex matrix pencils that are structurally stable. It is shown that there are no pairs $(M,N)$ of $m\times n$ and $n\times m$ complex matrices ($m,n\ge 1$) that are structurally stable under the contragredient equivalence $(S^{-1}MR, R^{-1}NS),$ in which $S$ and $R$ are nonsingular.

### Hardy’s and Miyachi’s theorems for the first Hankel – Clifford transform

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 710-715

UDC 517.5

We present an analog of Hardy's and Miyachi's theorems for the first Hankel$\,--\,$,Clifford transform.

### Concave shells of continuity modules

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 5. - pp. 716-720

UDC 517.9

The inequality
$$
\overline{\omega}(t)\leq\inf_{s>0}\left(\omega\left(\dfrac{s}{2}\right)+\dfrac{\omega(s)}{s}t\right)
$$
is proved, where $\omega(t)$ is a function of the modulus of continuity type and $\overline{\omega}(t)$ is its smallest concave majorant. The consequences obtained for Jackson's inequalities in $C_{2\pi}$ are presented.