A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation ${\Delta }^{p}f=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:\Omega \to {ℂ}^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f=(f₁,...,f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings f from...

We present a survey of the Lusin condition (N) for $W^{1,n}$-Sobolev mappings $f:G\to {ℝ}^n$ defined in a domain G of ${ℝ}^n$. Applications to the boundary behavior of conformal mappings are discussed.

We obtain Liouville type theorems for mappings with bounded $s$-distorsion between Riemannian manifolds. Besides these mappings, we introduce and study a new class, which we call mappings with bounded $q$-codistorsion.

Let $D$ be a nonempty open set in a metric space $(X,d)$ with $\partial D\ne \varnothing$. Define $$h_{D,c}(x,y)=log\left(1+c\frac{d(x,y)}{\sqrt{d_D\left(x\right)d_D\left(y\right)}}\right),$$ where $d_D\left(x\right)=d(x,\partial D)$ is the distance from $x$ to the boundary of $D$. For every $c\ge 2$, $h_{D,c}$ is a metric. We study the sharp Lipschitz constants for the metric $h_{D,c}$ under Möbius transformations of the unit ball, the upper half space, and the punctured unit ball.

Let f(z,t) be a Loewner chain on the Euclidean unit ball B in ℂⁿ. Assume that f(z) = f(z,0) is quasiconformal. We give a sufficient condition for f to extend to a quasiconformal homeomorphism of ${ℝ}^{2n}$ onto itself.