Vol 40, No 9 (2019)

Let \(\mathcal{M}\) be the class of analytic functions in the unit disk \(\mathbb{D}\) with the normalization f(0) = f′(0) − 1 = 0, and satisfying the condition

Functions in \(\mathcal{M}\) are known to be univalent in \(\mathbb{D}\). In this paper, it is shown that the harmonic mean of two functions in \(\mathcal{M}\) are closed, that is, it belongs again to \(\mathcal{M}\). This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in \(\mathcal{M}\) are shown to be starlike in \(\mathbb{D}\). However we conjecture that functions in \(\mathcal{M}\) are not necessarily starlike, as apparently supported by other examples.

Let P, Q be projections on a Hilbert space. We prove the equivalence of the following conditions: (i) PQ + QP ≤ 2(QPQ)^p for some number 0 < p ≤ 1; (ii) PQ is paranormal; (iii) PQ is M*-paranormal; (iv) PQ = QP. This allows us to obtain the commutativity criterion for a von Neumann algebra. For a positive normal functional φ on von Neumann algebra \(\mathcal{M}\) it is proved the equivalence of the following conditions: (i) φ is tracial; (ii) φ(PQ + QP) ≤ 2φ((QPQ)^p) for all projections P,Q ∈ \(\mathcal{M}\) and for some p = p(P, Q) ∈ (0,1]; (iii) φ(PQP) ≤ φ(P)^1/pφ(Q)^1/q for all projections P, Q ∈ \(\mathcal{M}\) and some positive numbers p = p(P, Q), q = q(P, Q) with 1/p+ 1/q = 1, p ≠ 2. Corollary: for a positive normal functional φ on \(\mathcal{M}\) the following conditions are equivalent: (i) φ is tracial; (ii) φ(A + A*) ≤ 2φ(∣A*∣) for all A ∈ \(\mathcal{M}\).

We consider the properties of biorthogonal systems induced by a convolution operator with Carleman kernel for a regular triangle. This is a perturbed singular operator with fixed singularities. We describe its set of anti-invariant points. To this end, we regularize the operator using a Carleman linear convolution shift that maps each triangle side to itself and changes its orientation, with the middle points of the sides being the fixed points of the shift. We search for a solution in the form of a Cauchy-type integral with unknown density. For this, both the theory of the Carleman boundary-value problem and the method of locally conformal gluing are used in an essential manner. We also apply the theory of elliptic functions that are generated by the corresponding doubly periodic group determined by the triangle as ‘half’ of the fundamental set. Using the method of contracting mappings in a Banach space, we study the corresponding homogeneous Fredholm integral equation of the second kind with regard to its solvability. Its fundamental system of solutions contains a single function; the fundamental system of solutions of the conjugated equation contains only the constant function. This makes it possible to use this equation for the construction of a system of biorthogonally conjugated analytic functions. More precisely, we consider a system of successive derivatives of a certain rational function determined by the Carleman kernel for the triangle and investigate the approximating properties of this system, as well as those of the corresponding biorthogonally conjugated system. This is a system of Cauchy-type integrals over the triangle boundary with a density which is invariant under the considered Carleman shift. Nontrivial decompositions of zero are obtained using the system of successive derivatives of the given rational function. The results are applied to the representation of some classes of analytical functions by means of the corresponding biorthogonal series.

We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic mappings. This solves also a recent problem of Dorff and Muir. In several of the cases, we illustrate mappings together with their minimal surfaces pictorially with the help of Mathematica software.

In this article we derive new estimates for the moduli of the Taylor coefficients of Bloch functions. We use one of these estimates to prove an inequality of an area type for such functions.

In this paper lower bounds for entire functions of exponential type and regular growth, zero sets of which have zero condensation indices, are obtained. In this case, the exceptional set consists of pairwise disjoint disks centered at zeroes. Sufficient conditions for radii of these circles are indicated. We also obtain a result on representation of analytic functions in the closure of a bounded convex domain (as well as analytic functions in domain and continuous up to the boundary) by series of exponential monomials. This result extends the classical result of A.F. Leont’ev to the case of multiple zero set of entire function. The obtained result is applied to the problem on distribution of singular points of a sum of series of exponential monomials at the boundary of its convergence domain.

Exact estimations of dimensions of spaces of bounded solutions of stationary Schrodinger equation with finite Dirichlet integral in terms of massive sets are obtained. It is proved that dimension of spaces of bounded solutions of this equation is not less than number of disjoint qD-massive subsets of manifold. This paper partly extends, the results of A.A. Grigor’yan, A.G. Losev (2017).

Hardy type inequalities with an additional nonnegative-term are established for compactly supported smooth functions on arbitrary open subsets and on convex domains of the Euclidean space. We prove Hardy-type inequalities in spatial domains with finite inner radius. Weight functions depend on the distance function to the boundary of the domain. We obtain one-dimensional L₁-inequalities. In particular cases we obtained sharp constants. Also new Hardy type inequality with remainders for the Riemann-Liouville fractional integrals is proved.

The infinite-dimensional complex Frechet manifolds \({\cal R}: = {\rm{Dif}}{{\rm{f}}_ + }({S^1})/{S^1}\) and \({\cal S}: = {\rm{Dif}}{{\rm{f}}_ + }({S^1})/{\rm{M\ddot ob}}({S^1})\) are the quotients of the group Diff₊(S¹) of orientation-preserving diffeomorphisms of the unit circle S¹ modulo subgroups of rotations and fractional-linear transformations respectively. These manifolds are the coadjoint orbits of the Virasoro group and the only ones having a Kähler structure. It motivates the study of their complex geometry. These manifolds are also closely related to string theory because they can be realized as the spaces of complex structures on loop spaces.

We consider Bernoulli distribution algebras, i.e. sets of distributions that are closed under transformations achieved by substituting independent random variables for arguments of Boolean functions from a given system. We establish that, unless the transforming set contains only essentially unary functions, the set of algebra limit points is either empty, single-element or no less than countable.