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S. Nasyrov

Publications and source records attributed to S. Nasyrov.

7 recordsLinked to original sources
arXiv2026

Distortion of the triangular ratio metric under Moebius transformations

Let $\mathbb{U}$ be the unit disk in the complex plane. Denote by $s_\mathbb{U}(x,y)$ the triangular ratio metric in $\mathbb{U}$; the value of $s_\mathbb{U}(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ to the value $\inf_{z\in \partial \mathbb{U}}(|x-z|+|z-y|)$. In the monograph by P.~Hariri, R.~Klén, and M.~Vuorinen "Conformally invariant metrics and quasiconformal mappings" (2020) the following problem was stated: for every Moebius automorphism of the unit disk, $w=f(z)=\frac{z+a}{1+za}$, $0\le a<1$, and every points $z_1$, $z_2\in \mathbb{U}$ the sharp inequality $s_\mathbb{U}(f(z_1),f(z_2))\le (1+a)s_\mathbb{U}(z_1,z_2)$ holds. We prove that the conjecture is valid.

math.CV↗
arXiv2026

Comparison of hyperbolic metric and triangular ratio metric in a square

Let $K$ be a square in the plane and $ρ_K(x,y)$ be the hyperbolic distance between $x$, $y\in K$. Denote by $s_K(x,y)$ the triangular ratio metric in $K$; for $x\neq y$ the value of $s_K(x,y)$ equals the ratio of the Euclidean distance $|x-y|$ between $x$, $y\in K$ to the value $\inf_{z\in \partial K}(|x-z|+|z-y|)$. We obtain a sharp estimate for the ratio of $þ(ρ_K(x,y)/2)$ to $s_K(x,y)$.

math.CV↗
arXiv2023

One Parameter Families of Conformal Mappings of Bounded Doubly Connected Polygonal Domains

We suggest an approximate method of finding a conformal mapping of an annulus onto an arbitrary bounded doubly connected polygonal domain. The method is based on the parametric Loewner--Komatu method. We consider smooth one parameter families $\mathcal{F}(z,t)$ of conformal mappings of concentric annuli onto doubly connected polygonal domains $\mathcal{D}(t)$ which are obtained from a fixed doubly connected polygonal domain $\mathcal{D}$ by making a finite number of rectilinear slits of variable lengths; meanwhile we do not require the family of domains $\mathcal{D}(t)$ to be monotonous. The integral representation for the conformal mappings $\mathcal{F}(z,t)$ has unknown (accessory) parameters. We find a PDE for this family and then deduce a system of PDEs describing dynamics of accessory parameters and conformal modulus of $\mathcal{D}(t)$ when changing the parameter $t$. We note that the right-hand sides of equations in the system of ODEs involve functions standing for speeds of movement of the end points of the slits. This allows us to control completely dynamics of the slits and to seek their agreed change, if we have more than one slit. We also give some results of numerical calculations which show the efficiency of the suggested method.

math.CV↗
arXiv2023

One-parameter families of conformal mappings of the half-plane onto polygonal domains with several slits

Among various methods of finding accessory parameters in the Schwarz-Christoffel integrals, Kufarev's method, based on the Loewner differential equation, plays an important role. It is used for describing one-parameter families of functions that conformally map a canonical domain onto a polygon with a slit the endpoint of which moves along a polygonal line starting from a boundary point. We present a modification of Kufarev's method for the case of several slits, the lengths of which have depend of each other in a certain way. We justify the method and find a system of ODEs describing the dynamics of accessory parameters. We also present the results of numerical calculations which confirm the efficiency of our method.

math.CV↗
arXiv2022

Intrinsic metrics in polygonal domains

We study inequalities between the hyperbolic metric and intrinsic metrics in convex polygonal domains in the complex plane. Special attention is paid to the triangular ratio metric in rectangles. A local study leads to an investigation of the relationship between the conformal radius at an arbitrary point of a planar domain and the distance of the point to the boundary.

math.CV↗