Spherically meanp-valent quasiregular mappings

We introduce spherically meanp-valent quasiregular mappings. Using the method of modulus of path families we prove a distortion theorem and describe the boundary behaviour of this class of mappings.     

Slow escaping points of quasiregular mappings

This article concerns the iteration of quasiregular mappings on \(\mathbb {R}^d\) and entire functions on \(\mathbb {C}\). It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let \(f:\mathbb {R}^d\rightarrow \mathbb {R}^d\) be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of f contains points at which the iterates \(f^n\) tend to infinity arbitrarily slowly. We also prove that, for any large R, there is a point x with modulus approximately R such that the growth of \(|f^n(x)|\) is asymptotic to the iterated maximum modulus \(M^{n}(R,f)\).     

Estimation of dilatations for mappings more general than quasiregular mappings

We consider the so-called ring Q-mappings, which are natural generalizations of quasiregular mappings in a sense of V?is?l?’s geometric definition of moduli. It is shown that, under the condition of nondegeneracy of these mappings, their inner dilatation is majorized by a function Q(x) to within a constant depending solely on the dimension of the space.     

On quasiregular mappings between smooth Jordan domains

It is proved that every proper quasiregular C² mapping w between two plane Jordan domains Ω∈C1,α and G∈C2,α, 0<α?1, satisfying the differential inequality |Δw|?M²|∇w|+N is Lipschitz continuous. This extends the main result of the author and M. Mateljevi? (Kalaj and Mateljevi?, 2006 [7]).     

The oscillation of harmonic and quasiregular mappings

Abstract. If u(z) is harmonic in with and we set A result is obtained which shows, in particular that if and then a bound for can be obtained in terms of for a suitable constant , so that the logarithm of the oscillation has an approximate convexity property. The proof uses classical inequalities of Hadamard and Borel–Carathéodory and this suggests a generalization to quasiregular mappings in . Such results are obtained, though necessarily in a less precise form because of the lack of good explicit estimates for -harmonic measures in spherical ring domains. Received: 9 November 2000 / Published online: 18 January 2002     

Subharmonicity of the modulus of quasiregular harmonic mappings

In this note we determine all numbers q∈R such that q|u| is a subharmonic function, provided that u is a K-quasiregular harmonic mappings in an open subset Ω of the Euclidean space Rn.     

On Dyakonov type theorems for harmonic quasiregular mappings

We prove two Dyakonov type theorems which relate the modulus of continuity of a function on the unit disc with the modulus of continuity of its absolute value. The methods we use are quite elementary, they cover the case of functions which are quasiregular and harmonic, briefly hqr, in the unit disc.     

Fixed points and normal families of quasiregular mappings

LetF be a family of mappingsK-quasiregular in some domainG. We show that if for eachf∈F, there existsk>1 such that thek-th iteratef ^k off has no fixed point, thenF is normal. Moreover, we examine to what extent this result holds if we consider only repelling fixed points, rather than fixed points in general. We also prove thatF is quasinormal, ifF contains only quasiregular mappings that do not have periodic points of some period greater than one inG. This implies that a quasiregular mappingf:ℝ ⁿ with an essential singularity in ∞ has infinitely many periodic points of any period greater than one. These results generalize results of M. Essén, S. Wu, D. Bargmann and W. Bergweiler for holomorphic functions.     

Remarks on the local index of quasiregular mappings

Journal d'Analyse Mathématique -     

Bloch radius, normal families and quasiregular mappings

Bloch's Theorem is extended to -quasiregular maps , where is the standard -dimensional sphere. An example shows that Bloch's constant actually depends on for .

     

A big Picard theorem for quasiregular mappings into manifolds with many ends

We study quasiregular mappings from a punctured Euclidean ball into -manifolds with many ends and prove, by using Harnack's inequality, a version of the big Picard theorem.

     

An egg-yolk principle and exponential integrability for quasiregular mappings

Quasiregular mappings f:ⁿⁿ are a natural generalization of analyticfunctions from complex analysis and provide a theory which isrich with new phenomena. In this paper we extend a well-knownresult of Chang and Marshall on exponential integrability ofanalytic functions in the disk, to the case of quasiregularmappings defined in the unit ball of ⁿ. To this end, an ‘egg-yolk’principle is first established for such maps, which extendsa recent result of the first author. Our work leaves open aninteresting problem regarding n-harmonic functions.     

A short proof of the self-improving regularity of quasiregular mappings

We provide a short proof of a theorem, due to Iwaniec and Martin (1993) and Iwaniec (1992), on the self-improving integrability of quasiregular mappings.