# Volume 61, № 10, 2009

### Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space

Babenko V. F., Bilichenko R. O.

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1299-1305

The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the $L_2$-norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space.

### Schur convexity and Schur multiplicative convexity for a class of symmetric functions with applications

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1306-1318

For $x = (x_1, x_2, …, x_n) ∈ (0, 1 ]^n$ and $r ∈ \{ 1, 2, … , n\}$, a symmetric function $F_n(x, r)$ is defined by the relation $$F_n(x,r) = F_n(x_1, x_2, …, x_n; r) = ∑_{1 ⩽ i_1 < i_2…i_r ⩽n } ∏^r_{j=1}\frac{1−x_{i_j}}{x_{i_j}},$$ where $i_1 , i_2 , ... , i_n$ are positive integers. This paper deals with the Schur convexity and Schur multiplicative convexity of $F_n(x, r)$. As applications, some inequalities are established by using the theory of majorization.

### Betweenness relation and isometric imbeddings of metric spaces

Dordovskii D. V., Dovgoshei A. A.

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1319-1328

We give an elementary proof of the classical Menger result according to which any metric space *X* that consists of more than four points is isometrically imbedded into \( \mathbb{R} \) if every three-point subspace of *X* is isometrically imbedded into \( \mathbb{R} \). A series of corollaries of this theorem is obtained. We establish new criteria for finite metric spaces to be isometrically imbedded into \( \mathbb{R} \).

### On extension of some generalizations of quasiconformal mappings to a boundary

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1329-1337

This work is devoted to the investigation of ring $Q$-homeomorphisms. We formulate conditions for a function $Q(x)$ and the boundary of a domain under which every ring $Q$-homeomorphism admits a homeomorphic extension to the boundary. For an arbitrary ring $Q$-homeomorphism $f: D → D’$ with $Q ∈ L_1(D)$; we study the problem of the extension of inverse mappings to the boundary. It is proved that an isolated singularity is removable for ring $Q$-homeomorphisms if $Q$ has finite mean oscillation at a point.

### On reduction of block matrices in a Hilbert space

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1338-1347

We study the problem of the reduction of self-adjoint block matrices $B = (B_ij)$ with given graph by a group of unitary block diagonal matrices. Under the condition that the matrices $B^2$ and $B^4$ are orthoscalar, we describe the graphs of block matrices for which this problem is a problem of *-finite, *-tame, or *-wild representation type.

### Trigonometric and orthoprojection widths of classes of periodic functions of many variables

Romanyuk A. S., Romanyuk V. S.

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1348-1366

We obtain exact order estimates for trigonometric and orthoprojection widths of the Besov classes $B^r_{p,θ}$ and Nikol’skii classes $Hr p$ of periodic functions of many variables in the space $L_q$ for certain relations between the parameters $p$ and $q$.

### On the integral characterization of some generalized quasiregular mappings and the significance of the conditions of divergence of integrals in the geometric theory of functions

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1367-1380

The paper deals with the theory of space mappings. For a generalization of quasiregular mappings important for the investigation of the Sobolev and other known classes of mappings, we propose a simple condition satisfied by all mappings of this kind and only by these mappings. On the basis of conditions of divergence of the integrals, we establish sufficient conditions for the normality of the families of the analyzed classes of mappings and solve the problem of removing isolated singularities. Some applications of the obtained results to mappings from the Sobolev class are discussed.

### On some generalizations of nearly normal subgroups

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1381-1395

A subgroup $H$ of a group $G$ is called almost polycyclically close to a normal group (in $G$) if $H$ contains a subgroup $L$ normal in $H^G$ for which the quotient group $H^G /L$ is almost polycyclic. The group G is called an anti-$PC$-group if each its subgroup, which is not almost polycyclic, is almost polycyclically close to normal. The structure of minimax anti-$PC$-groups is investigated.

### Normal and tangential geodesic deformations of the surfaces of revolution

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1396-1402

We study special infinitesimal geodesic deformations of the surfaces of revolution in the Euclidean space $E^3$.

### Order equalities for some functionals and their application to the estimation of the best $n$-term approximations and widths

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1403-1423

We study the behavior of functionals of the form $\sup_{l>n} (l-n)\left(∑^l_{k=1} \frac1{ψ^r(k)} \right)^{−1/r}$, where $ψ$ is a positive function, as $n → ∞$: The obtained results are used to establish the exact order equalities (as $n → ∞$) for the best $n$-term approximations of $q$-ellipsoids in metrics of the spaces $S^p_{φ}$: We also consider the applications of the obtained results to the determination of the exact orders of the Kolmogorov widths of octahedra in the Hilbert space.

### Unitarization of representations of a partially ordered set associated with a graph $\tilde{E}_6$

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1424-1433

It is shown that any Schur representation of a poset associated with a graph $\tilde{E}_6$ can be unitarized with a certain character. The description of characters for which it is possible to unitarize the Schur representations of $\tilde{E}_6$ is presented.

### Boundedness of multilinear singular integral operators on the homogeneous Morrey–Herz spaces

Ukr. Mat. Zh. - 2009. - 61, № 10. - pp. 1434-1440

A boundedness result is established for multilinear singular integral operators on the homogeneous Morrey–Herz spaces. As applications, two corollaries about interesting cases of the boundedness of the considered operators on the homogeneous Morrey–Herz spaces are obtained.