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Given a homeomorphism f:X→Yf:X\to Y between QQ-dimensional spaces X,YX,Y, we show that ff satisfying the metric definition of quasiconformality outside suitable exceptional sets implies that ff belongs to the Sobolev class Nloc1,p(X;Y){N}_{{\rm{loc}}}^{1,p}\left(X;\hspace{0.33em}Y), where 1≤p≤Q1\le p\le Q, and also implies one direction of the geometric definition of quasiconformality. Unlike previous results, we only assume a pointwise version of Ahlfors QQ-regularity, which in particular enables various weighted spaces to be included in the theory. Notably, even in the classical Euclidean setting, we are able to obtain new results using this approach. In particular, in spaces including the Carnot groups, we are able to prove the Sobolev regularity f∈Nloc1,Q(X;Y)f\in {N}_{{\rm{loc}}}^{1,Q}\left(X;\hspace{0.33em}Y) without the strong assumption of the infinitesimal distortion hf{h}_{f} belonging to L∞(X){L}^{\infty }\left(X).

On the Riemann sphere, we consider the ptolemaic characteristic of a four of non-empty pairwise non-intersecting compact subsets (generalized tetrad, or generalized angle). We obtain an estimate for distortion of this characteristic under the inverse to a K-quasimeromorphic mapping of the Riemann sphere which takes each of its values at no more then N different points. The distortion function in this estimate depends only on K and N. In the case K=1, it is an essentially new property of complex rational functions.

In an attempt to develop higher-dimensional quasiconformal mappings on metric measure spaces with curvature conditions, i.e. from Ahlfors to Alexandrov, we show that for n≥2 a noncollapsed RCD(0,n) space with Euclidean volume growth is an n-Loewner space and satisfies the infinitesimal-to-global principle.

In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region.

In this paper, we study the growth of the mth derivative of an arbitrary algebraic polynomial in bounded and unbounded general domains of the complex plane in weighted Lebesgue spaces. Further, we obtain estimates for the derivatives at the closure of this regions. As a result, estimates for derivatives on the entire complex plane were found.

In this paper, three equivalent conditions of ρ-harmonic Teichmüller mapping are given firstly. As an application, we investigate the relationship between a ρ-harmonic Teichmüller mapping and its associated holomorphic quadratic differential and obtain a relatively simple method to prove Theorem 2.1 in [1]. Furthermore, the representation theorem of 1/|ω|2-harmonic Teichmüller mappings is given as a by-product. Our results extend the corresponding researches of harmonic Teichmüller mappings.

In this paper, we study Bernstein-Walsh type estimates for the higher-order derivatives of an arbitrary algebraic polynomial on quasidisks.

We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

Abstract In this paper we study the hyperbolic and Euclidean area distortion of measurable sets under some classes of K-quasiconformal mappings from the upper half-plane and the unit disk onto themselves, respectively.

A numerical method of quasiconformal mappings for solving the coefficient problems of finding eigenvalues of the conductivity tensor having information about its directions in an anisotropic medium using applied quasipotential tomographic data is generalized. The corresponding algorithm is based on the alternate solving of problems on quasiconformal mappings and parameter identification. The results of numerical experiments of imitative restoration of environment structure are presented.

Conventional splines offer powerful means for modeling surfaces and volumes in three-dimensional Euclidean space. A one-dimensional quaternion spline has been applied for animation purpose, where the splines are defined to model a one-dimensional submanifold in the three-dimensional Lie group. Given two surfaces, all of the diffeomorphisms between them form an infinite dimensional manifold, the so-called diffeomorphism space. In this work, we propose a novel scheme to model finite dimensional submanifolds in the diffeomorphism space by generalizing conventional splines. According to quasiconformal geometry theorem, each diffeomorphism determines a Beltrami differential on the source surface. Inversely, the diffeomorphism is determined by its Beltrami differential with normalization conditions. Therefore, the diffeomorphism space has one-to-one correspondence to the space of a special differential form. The convex combination of Beltrami differentials is still a Beltrami differential. Therefore, the conventional spline scheme can be generalized to the Beltrami differential space and, consequently, to the diffeomorphism space. Our experiments demonstrate the efficiency and efficacy of diffeomorphism splines. The diffeomorphism spline has many potential applications, such as surface registration, tracking and animation.

Let S denote the set of all functions f which are analytic and univalent in the unit disk D normalized so that f(z)=z+a2z2+…. Let S∗ and C be those functions f in S for which f(D) is starlike and convex, respectively. For 0≤k