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We establish that Grunsky norm of any normalized univalent function on a quasidisk is completely determined by the squares of holomorphic abelian differentials (in contrast to Teichmüller norm, which relates to all integrable holomorphic quadratic differentials). The result has important interesting applications in several fields of geometric function theory: it provides an explicit representation of Fredholm eigevalues of all quasiconformal curves, sheds a new light on the distortion theory for univalent functions with quasiconformal extension, etc. [ABSTRACT FROM AUTHOR]

In this note, we present the state-of-the-art theory of bi-Sobolev mappings. We recall that f is a Sobolev homeomorphism if f belongs to W loc 1 , 1 ∩ Hom (Ω ; Ω ′) , and f is a bi-Sobolev map if and only if f and f - 1 are Sobolev homeomorphisms. This concept, introduced in Hencl et al. (J. Math. Anal. Appl. 355, 22–32 2009), plays a central role in Geometric Function Theory. For instance, we just mention here that maps of bi-Sobolev type are strictly related to the notion of mappings of finite distortion; see, among others, the papers (Hencl and Koskela 2014) Pratelli (Nonlinear Anal. 154, 258–268 2017). [ABSTRACT FROM AUTHOR]

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk 𝔻 is due to the dissertation of Luzin. Later on, the known monograph of Vekua was devoted to boundary-value problems only with Hölder continuous data for generalized analytic functions, i.e., continuous complex-valued functions f(z) of the complex variable z = x + iy with generalized first partial derivatives by Sobolev satisfying equations of the form ∂ z ¯ f + af + b f ¯ = c , where the complexvalued functions a; b, and c are assumed to belong to the class Lp with some p > 2 in smooth enough domains D in ℂ. Our last paper [12] contained theorems on the existence of nonclassical solutions of the Hilbert boundaryvalue problem with arbitrary measurable data (with respect to logarithmic capacity) for generalized analytic functions f : D → ℂ such that ∂ z ¯ f = g with the real-valued sources. On this basis, the corresponding existence theorems were established for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations △U = G ∈ Lp; p > 2, with arbitrary measurable boundary data over logarithmic capacity. The present paper is a natural continuation of the article [12] and includes, in particular, theorems on the existence of solutions for the Hilbert boundary-value problem with arbitrary measurable data for the corresponding nonlinear equations of the Vekua type ∂ z ¯ f z = h z q f z . On this basis, existence theorems were also established for the Poincar´e boundary-value problem and, in particular, for the Neumann problem for the nonlinear Poisson equations of the form △U(z) = H(z)Q(U(z)) with arbitrary measurable boundary data over logarithmic capacity. The Dirichlet problem was investigated by us for the given equations, too. Our approach is based on the interpretation of boundary values in the sense of angular (along nontangential paths) limits that are a conventional tool of the geometric function theory. As consequences, we give applications to some concrete semi-linear equations of mathematical physics arising from modelling various physical processes. Those results can also be applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media. [ABSTRACT FROM AUTHOR]

Recently the author has presented a new approach to solving the coefficient problems for holomorphic functions based on the deep features of Teichmüller spaces. It involves the Bers isomorphism theorem for Teichmüller spaces of punctured Riemann surfaces. The aim of the present paper is to provide new applications of this approach and extend the indicated results to more general classes of functions [ABSTRACT FROM AUTHOR]

In the geometric theory of functions of a complex variable, the well-known direction is related to the estimates of the products of the inner radii of pairwise nonoverlapping domains. This direction is called extreme problems in classes of pairwise nonoverlapping domains. One of the problems of this type is considered in the present work. [ABSTRACT FROM AUTHOR]

In this paper, we describe the inhomogeneous Hilbert boundary-value problem of the theory of analytic functions with an infinite index and a boundary condition for a half-plane. The coefficients of the boundary condition are Hölder-continuous everywhere except for a finite number of singular points at which the argument of the coefficient function has second-type discontinuities (of a power order with exponent <1). We obtain formulas for the general solution of the inhomogeneous problem and discuss the existence and uniqueness of the solution. The study is based on the theory of entire functions and the geometric theory of functions of a complex variable. [ABSTRACT FROM AUTHOR]

We consider the well-known problem of the geometric theory of functions of a complex variable on non-overlapping domains with free poles on radial systems. The main results of the present work strengthen and generalize several known results for this problem. [ABSTRACT FROM AUTHOR]

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional [ABSTRACT FROM AUTHOR]

Some extremal problems of the geometric theory of functions of a complex variable related to the estimates of functionals defined on systems of non-overlapping domains are considered. Till now, many such problems have not been solved, though some partial solutions are available. In the paper, the improved method is proposed for solving the problems on extremal decomposition of the complex plane. The main results generalize and strengthen some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane. [ABSTRACT FROM AUTHOR]

We review the main results of the geometric theory of ordinary differential equations obtained by the prominent Russian geometer N. V. Stepanov (1926–1991). Some results obtained by Stepanov are illustrated by examples of third- and five-order equations. [ABSTRACT FROM AUTHOR]

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin's paper in the journal "Russian Mathematical Surveys" in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of the continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on recent progress in the geometric theory of spiral-like functions. We prove convergence theorems and illustrate them by numerical simulations. [ABSTRACT FROM AUTHOR]

Results and applications of the method of modules in geometric function theory are presented. The method was originally created by J. A. Jenkins, and further developed in works of the Leningrad-St. Petersburg mathematical school. A retrospective description of the origin of the method is given, and the determining role of Jenkins in the development of the method of the extremal metric is pointed out. [ABSTRACT FROM AUTHOR]

The classics of function theory (E. Borel, H. Lebesgue, R. Baire, W. H. Young, F. Hausdorff, et al.) have laid down the foundation of the classical descriptive theory of functions. Its initial notions are the notions of a descriptive space and of a measurable function on it. Measurable functions were defined in the classical preimage language. However, a specific range of tasks in theory of functions, measure theory, and integration theory emergent on this base necessitates the usage of the entirely different postclassical cover language, equivalent to the preimage language in the classical case. By means of the cover language, the general notions of a prescriptive space and distributable and uniform functions on it are introduced in this paper and their basic properties are studied. [ABSTRACT FROM AUTHOR]

Within the geometric theory of functions, we study one of the classical problems of extreme decomposition of a complex plane. [ABSTRACT FROM AUTHOR]

Applications of geometric function theory to some inequalities for algebraic polynomials and rational functions normalized on circular arcs are considered. In particular, coefficients estimates and also covering and distortion theorems are proved. The latter theorems supplement recent results of the author and V. N. Dubinin. [ABSTRACT FROM AUTHOR]

The dependence of values of functionals occurring in some extremal decomposition problems on the location of the poles of the associated quadratic differentials is established. The proof is based on an analysis of geometric properties of trajectories of two quadratic differentials. [ABSTRACT FROM AUTHOR]

It is proved that the embedding problem with a cyclic kernel for number fields has a solution if 2 completely decomposes in the embeddable field and if the consistency condition holds. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

We give parametric expressions for extremals of the Grötzsch's problem. [ABSTRACT FROM AUTHOR]

We consider some examples of systems whose functionality can be described with the help of mathematical models based on singular integral equations with non-Carleman shift. For this type of equations, we have obtained an estimate for the dimension of the kernel of the corresponding operator and proposed numerical methods for both evaluating the dimension of the kernel and constructing solutions. [ABSTRACT FROM AUTHOR]

The classical Lindelöf principle on the behavior of Green’s function under a regular mapping is generalized to the case of Robin function with a pole at a boundary point. In addition, inequalities inverse to the Lindelöf principle are considered. As corollaries, certain analogs of Mityuk’s theorems on the behavior of the inner radius of a domain are established. Also a particular case of a Mityuk’s theorem and a Kloke’s result on the change of the condenser capacity under a multivalent mapping are supplemented. Bibliography: 19 titles. [ABSTRACT FROM AUTHOR]

Pseudo-characters of groups have recently found applications in the theory of classical knots and links in ℝ3. More precisely, there is a connection between pseudo-characters of Artin’s braid groups and properties of links represented by braids. In the present work, this connection is investigated and the notion of kernel pseudo-characters of braid groups is introduced. It is proved that a kernel pseudo-character ϕ and a braid β satisfy Ιϕ(β)І > Cϕ, where Cϕ is the defect of ϕ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudo-characters is studied and a way to obtain nontrivial kernel pseudo-characters from an arbitrary braid group pseudo-character that is not a homomorphisrn is described. This allows one to use an arbitrary nontrivial braid group pseudo-character for recognition of prime knots and links. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

It is proved that under appropriate assumptions, the solutions f of quadratic congruences Q( f) ≡ 0 (mod n), n ≤ x, with ( f| n) = ±1, are distributed asymptotically equally as x → ∞. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

Some problems on extremal decomposition in families of nonoverlapping domains containing systems of biangles with free vertices on a circle are considered. Simultaneously, some progress in solving the classical problem on the maximum of a well-known conformal invariant is achieved. This exhibits the role of symmetric configurations in extremal problems under consideration. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The capacity approach and symmetrization method arc adapted to some extremal decomposition problems on the unit disk or an annulus. The problems on the maximum product of the interior radii of pairwise nonoverlapping domains and the maximum product of the Robin radii of such domains are considered. New invariants with respect to the Möbius transformations of the Riemann sphere are introduced. In particular, for these invariants problems on extremal decomposition with free poles on the unit circle are investigated. Bibliography: 19 titles. [ABSTRACT FROM AUTHOR]

The problem of decomposition and composition of normal ring open sets is considered. The invariance of compacts generating such sets under quasiisometries and quasiconformal mappings is established under certain natural assumptions. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

Problems on extremal decomposition of a multiply connected domain are considered. Two theorems exhibiting the role in such problems of quadratic differentials that are perfect squares are proved. As an implication, it is shown that the extremal metric approach used in the paper naturally leads to the results recently obtained by Dubinin and Eyrikh (Zap. Nauchn. Semin. POMI, 314, 52–75 (2004)) with the use of the capacity theory for generalized condensers. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]

The article presents the main results concerning smooth semifree actions of finite groups on spheres. One shows that for every holomorphic function having an isolated singularity, there exists a smooth semifree action on a (possibly homotopic) sphere where the fixed-point set is the boundary for the singularity of the given function. [ABSTRACT FROM AUTHOR]

The paper presents a simple extremal metric approach to problems on extremal decomposition in families of systems of nonoverlapping domains of different types. The approach is based on using general properties of trajectories of the associated quadratic differentials for simpler extremal decomposition problems. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

New definitions of normal (in the sense of Grötzsch) annular and circular domains with radial slits are considered. Their geometric and functional properties are established. Arguments based on the method of extremal metric in the sense of Fuglede allow one to extend these properties to the n-dimensional case, n ≥ 2. The properties established provide a means for simplifying the proofs of the related results by Strebel and Reich and for bringing them to a complete form. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

The paper considers the problem on extremal decomposition of a multiply connected domain G ⊂ ℂ in the case where the associated quadratic differential is a perfect square. It is shown that in the case considered, the value of the functional for this extremal decomposition is the least one in a certain class of decompositions. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

Uniqueness theorems are proved for the water-wave problem involving either a body intersecting the free surface at arbitrary angles or a pair of symmetric plates floating in the free surface. Proofs combine conformal mappings and the Vainberg-Maz’ya identity. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

The article presents a study about a boundary-value problem with almost oscillatory spectrum. It demonstrates and proves that the case of the kernel function, which involves a Green function of the boundary-value problem, is sign-regular. Several theorems about these boundary-value problems are discussed.

We construct global generating functions of the initial and of the evolution Lagrangian submanifolds related to a Hamiltonian flow. These global parameterizations are realized by means of Amann—Conley—Zehnder reduction. In some cases, we have to to face generating functions that are weakly quadratic at infinity; more precisely, degeneracy points can occurs. Therefore, we develop a theory which allows us to treat possibly degenerate cases in order to define a Chaperon—Sikorav—Viterbo weak solution of a time-dependent Hamilton-Jacobi equation with a Cauchy condition given at time t = T ( T > 0). The starting motivation is to study some aspects of Mayer problems in optimal control theory. [ABSTRACT FROM AUTHOR]

We study conformal mappings of a band without some vertical symmetric slits into the plane without horizontal slits (that lie on the real axis). Sharp two-sided estimates involving the lengths of slits and other parameters are obtained. In terms of the spectral theory of Jacobi matrices, we obtain sharp estimates involving isospectral parameters and norms of potentials. Bibliography: 9 titles. [ABSTRACT FROM AUTHOR]

Properties of the reduced module of a half-strip are investigated. The half-strip is considered as a triangle with a vertex at infinity. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

New applications of generalized condensers with two or more plates are considered. Based on one and the same approach and simplest properties of such condensers, boundary distortion theorems for analytic univalent functions bounded in a disk are established, as well as bounds for certain combinations of coefficients in expansions of such functions, inequalities for polynomials, and theorems on extremal decompositions of the complex sphere. Part of the results obtained contains new information on the Schwarzian derivative of a regular univalent function. Bibliography: 21 titles. [ABSTRACT FROM AUTHOR]

The present paper continues the series of articles published in “Zapiski Nauchnykh Seminarov POMI” and devoted to the problems indicated in the title and related questions. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

For analytic functions satisfyingMeyman's majorization conditions, with the use of classical properties of conformal mappings exact inequalities complementing and strengthening the results of Akhiezer and Meyman are derived. As a corollary, for the modulus of the derivative of a rationally-trigonometric function a Bernstein type bound, which implies the result by Borwein, Erdelyi, and Zhang, is obtained. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

We study classical ensembles of sample covariance matrices introduced by Wishart. We discuss Stein's method for the asymptotic approximation of expectations of functions of the normalized eigenvalue counting measure for high-dimensional matrices. The method is based on a differential equation for the density of the Marchenko-Pastur law. Bibliography: 19 titles. [ABSTRACT FROM AUTHOR]

Examines the quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. Abstract scheme for obtaining higher-order conditions in smooth extremal problems with constraints; Creation of the quadratic theory of conditions for Pontryagin minimum in the general problem of the calculus of variations that does not contain the local relation g(x,u,t) = 0; Decoding of the basic constant for the set of so-called "Pontryagin sequences".

The classical Schwarz inequality on the boundary of the disk for the functions regular in the disk is refined in various directions. Inequalities involving zeros of the function, an inequality for points mapped to symmetric points on the circle, and an inverse estimate for univalent functions are presented. Other inequalities are discussed, and the possibility of applying them to estimates of polynomials and rational functions is indicated. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

A sharp estimate of the product ... Rαk² (Dk, bk) (as usual, R(D, b) denotes the conformal radius of a domain D with respect to a point b ∈ D) in the family of all quadruples of nonoverlapping simply connected domains {Dk}, bk ∈ Dk, k = 1,...,4, is obtained. Here, {b1,...,b4} are four arbitrary distinct points on &Cmacr;, α1 α2 = 1, α3 = α4 = α and α is an arbitrary positive number. The proof involves the solution of the problem on maximizing a certain conformal invariant, which is related to the problem under consideration. Bibliography: 5 titles. [ABSTRACT FROM AUTHOR]

Applications of the geometric theory of functions to inequalities for algebraic polynomials are considered. The main attention is paid to constructing a univalent conformal mapping for a given polynomial and to applying the Lebedev and Nehari theorems to this mapping. A new sharp inequality of Bernshtein type for polynomials with restrictions on the growth on a segment or on a circle, inequalities with restrictions on the zeros of the polynomial, and other inequalities are obtained. In particular, classical inequalities by Mar/coy, Bernshtein, and Schur are strengthened. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

Let ℜ be a finite Riemann surface. For a quadratic differential on ℜ associated with a certain problem on extremal decomposition of ℜ into n domains, a parametric family of quasiconformal mappings fΚ : ℜ → ℜΚ K = (k1,...,ks), ki ∈ C. is defined. These mappings map the domains of the extremal decomposition of ℜ onto the domains of the extremal decomposition of ℜΚ. This allows one to study the functional dependence of the problem on the parameters. Bibliography: 6 titles. [ABSTRACT FROM AUTHOR]