# Volume 70, № 7, 2018

### On the dependence of the norm of a multiply monotone function on the norms of its derivatives

Bondarenko A. R., Kovalenko O. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 867-875

We establish necessary and sufficient conditions for a system of positive numbers Mk1 , Mk2 , Mk3 , Mk4 , 0 = k1 < < k2 < k3 \leq r 3, k4 = r guaranteeing the existence of an (r 2)-monotone function x on the half line such that \| x(ki)\| \infty = Mki , i = 1, 2, 3, 4.

### On boundary values of three-harmonic Poisson integral on the boundary of a unit disk

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 876-884

Let $C_0$ be a curve in a disk $D = \{ | z| < 1\}$ tangential to a circle at the point $z = 1$ and let $C_{\theta}$ be the result of rotation of this curve about the origin $z = 0$ by an angle \theta . We construct a bounded function $u(z)$ three-harmonic in $D$ with zero normal derivatives $\cfrac{\partial u}{\partial n}$ and $\cfrac{\partial 2u}{\partial r_2}$ on the boundary such that the limit along $C_{\theta}$ does not exist for all $\theta , 0 \leq \theta \leq 2\pi $.

### Birosettes are model flexors

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 885-904

A new family of polyhedra called birosettes is presented. The geometric features of birosettes are analyzed. The model flexibility of birosettes is explained.

### Principally Goldie$\ast$ -lifting modules

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 905-912

A module $M$ is called а principal Goldie$\ast$ -lifting if, for every proper cyclic submodule $X$ of $M$, there is a direct summand $D$ of $M$ such that $X\beta \ast D$. We focus our attention on principally Goldie $\ast$ -lifting modules as a generalization of lifting modules. Various properties of these modules are presented.

### Subdivision of spectra for some lower triangular double-band matrices as operators on $c_0$

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 913-922

The generalized difference operator $\Delta_{a,b}$ was defined by El-Shabrawy: $\Delta{a,b}x = \Delta_{a,b} (x_n) = (a_nx_n + b_{n-1}x_{n-1})^{\infty}_{n = 0}$ with $x_1 = b_1 = 0$, where $(a_k), (b_k)$ are convergent sequences of nonzero real numbers satisfying certain conditions. We completely determine the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator $\Delta_{a,b}$ in the sequence space $c_0$.

### Entire functions of order zero with zeros on a logarithmic spiral

Tarasyuk S. I., Zabolotskii N. V., Zabolotskyi M. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 923-932

We prove the Valiron-type and Valiron – Titchmarsh-type theorems for entire functions of order zero with zeros on a logarithmic spiral.

### ORV sequences with nondegenerate groups of regular points

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 933-951

We define a class of ORV sequences with nondegenerate groups of regular points and consider some properties of this sequences.

### On convergence of mappings in metric spaces with direct and inverse modulus conditions

Sevost'yanov E. A., Skvortsov S. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 952-687

For mappings in metric spaces satisfying one inequality with respect to the modulus of families of curves, we establish the property of lightness of the limit mapping. It is shown that the uniform limit of these mappings is a light mapping, whenever the function responsible for the distortion of the families of curves, is of finite mean oscillation at every point. In addition, for one class of homeomorphisms of metric spaces, we prove theorems on the equicontinuity of the families of inverse mappings.

### Estimation of equimeasurable rearrangements in the anisotropic case

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 968-977

We study the classes of functions satisfying the reverse Holder inequality on segments in the multidimensional case. For these classes, we obtain sharp estimates of the “norms” of equimeasurable rearrangements.

### Entire functions share two half small functions

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 978-987

The paper generalizes a result by P. Li and C. C. Yang [Illinois J. Math. – 2000. – 44. – P. 349 – 362] and extends the previous work of G. Qiu [Kodai Math. J. – 2000. – 23. – P. 1 – 11].

### A note on strongly split Lie algebras

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 988-991

Split Lie algebras are maybe the most known examples of graded Lie algebras. Since an important category in the class of graded algebras is the category of strongly graded algebras, we introduce, in a natural way, the category of strongly split Lie algebras $L$ and show that if $L$ is centerless, then $L$ is the direct sum of split ideals each of which is a split-simple strongly split Lie algebra.

### Simpson-type inequalities for geometrically relative convex functions

Awan M. U., Noor K. I., Noor M. A.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 7. - pp. 992-1000

We consider a class of geometrically relative convex functions and deduce several new integral inequalities of Simpson’s type via geometrically relative convex functions. The ideas and techniques used in the paper may stimulate further research in this area.