Sergey Graf

• PhD
• Professor (Associate) at Tver State University

The article is devoted to the definition and properties of the class of diffeomorphisms ofthe unit disk D = { z : | z| < 1} on the complex plane C for which the harmonic measure of theboundary arcs of the slit disk has a limited distortion, i.e. is quasiinvariant. Estimates for derivativemappings of this class are obtained. We prove that such mappi...

The paper is devoted to the proof of new estimations of harmonic measure of a boundary arc of a Jordan domain D. The main result allows to estimate a distortion of harmonic measure under locally quasiconformal or quasiconformal mappings of the unit disk of complex plane onto domain D. As an application of the main results, some estimations for quas...

The paper is devoted to finding conditions, sufficient for uniform local univalence of sense-preserving mappings, harmonic in the unit disc of the complex plane; the conditions are given in terms of the generalized Schwarzian derivative introduced by R. Hernández and M. J. Martín. The main section contains proofs of the conditions of univalence and...

The plane domain D is called R-convex if D contains each compact set bounded by two shortest sub-arcs of the radius R with endpoints w1, w2 ∈ D, |w1−w2

In the paper we obtain some analogues of Nehari’s univalence conditions for sense-preserving functions that are harmonic in the unit disc D = {z ∈ C : |z| < 1}.

We obtain estimations of the pre-Schwarzian and Schwarzian derivatives in terms of the order of family in linear and affine invariant families L of sense preserving harmonic mappings of the unit disk D. As the converse result the order of family L is estimated in terms of suprema of Schwarzian and pre-Schwarzian norms over the family L. Main result...

For a locally univalent sense-preserving harmonic mapping $f=h+\overline{g}$ defined on the unit disk $\ID =\{z\in\mathbb C:\, |z|<1\}$, let $d_f(z)$ be the radius of the largest (univalent) disk on the manifold $f(\ID)$ centered at $f(z_0)$ ($|z_0|<1$). One of the aims of the present investigation is to obtain sharp upper and lower bounds for the...

Some of the earlier results of author concerning distortion of the moduli of ring domains under planar locally quasiconformal mappings are generalized on the case of locally quasiconformal mappings in Rn, n ≥ 2. The main result of the article represents the sharp double-sided estimation of modulus M(D) of the image D of the concentric spherical rin...

In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic $\Phi$-like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas'yanyuk \cite{Kas59} and...

We obtain sharp estimates for the module of functions in the classes of normalized locally quasiconformal authomorphisms of the unit disk with given majorants of M. A. Lavrent’ev’s characteristic. The estimates are analogs of Schwarz’s lemma and A. Mori’s theoremand they imply the classical growth theorems for quasiconformal authomorphisms of the d...

In linear- and affine-invariant families of harmonic mappings of the unit disk we prove some differential inequalities such as the sharp two-sided estimate of the Jacobian and an estimate of the curvature of the image of the circle. Key words and phrasesharmonic mappings-linear- and affine-invariant families of functions-order of a family

We give a final solution to the problem of the possibility of a finitely valent locally biholomorphic mapping from an arbitrary multiconnected domain on a complex plane onto the entire complex plane with the indication of the least valency constant.