Heinz‐type inequality and bi‐Lipschitz continuity for quasiconformal mappings satisfying inhomogeneous biharmonic equations

Abstract

Let Hom+false(double-struckTfalse) be the class of all sense‐preserving homeomorphic self‐mappings of double-struckT=false{z=x+iy∈double-struckC:false|zfalse|=1false}. The aim of this paper is twofold. First, we obtain Heinz‐type inequality for (K,K′)‐quasiconformal mappings satisfying inhomogeneous biharmonic equation Δ(Δω) = g in unit disk double-struckD with associated boundary value conditions normalΔω|T=φ∈scriptCfalse(double-struckTfalse) and ω|T=f∗∈Hom[+false(double-struckTfalse). Second, we establish biLipschitz continuity for (K,K′)‐quasiconformal mappings satisfying aforementioned inhomogeneous biharmonic equation when K′,false|false|φfalse||∞:=supz∈double-struckDfalse|φfalse(zfalse)false| and false|false|gfalse||∞:=supz∈double-struckDfalse|gfalse(zfalse)false| are small enough.

On asymptotically sharp bi-Lipschitz inequalities of quasiconformal mappings satisfying inhomogeneous polyharmonic equations

“…For example, the Lipschitz continuity of harmonic quasiconformal mappings has been discussed by many authors (see [22,24,30,36,40,42]). The Lipschitz continuity of (K, K ′ )quasiconformal harmonic mappings (see Section 1.2) has also been investigated in [7,26,48]. On the discussion of the related topic, we refer to [12,13,18,21,27,35,42,45,46] and the related references therein.…”

On a Quasi‐isometry for Quasiconformal Mapping Satisfying Poisson Differential Inequality

“…According to Lemma 2.2, we can get a Schwarz-Pick type inequality for mappings satisfying the Poisson's equation. This kind of lemma has been established in [25]. For the completeness, we will give the proof here.…”

“…The hyperbolically partial derivative on of a quasiconformal mapping f from domain D of hyperbolic type into domain of hyperbolic type is defined as In [19], Knežević and Mateljević obtained the following estimate on hyperbolically partial derivative of K ‐quasiconformal harmonic self‐mappings of . This result has been generalized in [4, 6, 25, 28]. Theorem A Let f be a K ‐quasiconformal harmonic self‐mapping of .…”

“…In [19], Knežević and Mateljević obtained the following estimate on hyperbolically partial derivative of K-quasiconformal harmonic self-mappings of D. This result has been generalized in [4,6,25,28].…”