Distortion of Quasiregular Mappings and Equivalent Norms on Lipschitz-Type Spaces

Abstract

We prove a quasiconformal analogue of Koebe’s theorem related to the average Jacobian and use a normal family argument here to prove a quasiregular analogue of this result in certain domains inn-dimensional space. As an application, we establish that Lipschitz-type properties are inherited by a quasiregular function from its modulo. We also prove some results of Hardy-Littlewood type for Lipschitz-type spaces in several dimensions, give the characterization of Lipschitz-type spaces for quasiregular mappings by the average Jacobian, and give a short review of the subject.

John Disks and K-Quasiconformal Harmonic Mappings

“…Proof of Theorem 5. Let f = h + g be a K-quasiconformal harmonic mapping of D onto a bounded domain D, where h and g are analytic in D. By [14,Proposition 13] and Theorem F, for r ∈ (0, 1) and all θ ∈ [0, 2π], we obtain…”

Bounds for Jacobian of harmonic injective mappings in n-dimensional space

“…152-153. The space version was communicated on International Conference on Complex Analysis and Related Topics (Xth Romanian-Finnish Seminar, August [14][15][16][17][18][19]2005, Cluj-Napoca, Romania), by Mateljević, cf. also [18].…”

“…Multiplying functions by constants less than 1 if necessary, we may assume, without loss of generality, that the boundary of the image f j (B n ) always contains a point on the unit spere, and thus use a normal family argument (since infinity and point on the unit sphere are on a fixed spherical distance) to pass to a convergent subsequence. Now note that because of the Gehring distortion property (see [21] p. 63, [17]), f j ( 1 2 B n ) will contain a ball around zero of fixed radius, so the limit cannot degenerate to a constant function. But then the limit, say g, is in the family.…”

Radial growth and Hardy-Littlewood-type theorems on hyperbolic harmonic functions

“…For the extensive discussions on this topics, see [1,13,[23][24][25]. For the extensive discussions on this topics, see [1,13,[23][24][25].…”