On locally quasiconformal mappings

A homeomorphism w=f(z) of a domain D is called a locally quasiconformal mapping, if for each subdomain D' of D with 'D, the restriction of f(z) on D' is a quasiconformal mapping. We give some conditions for a measurable function μ(z) on the unit disc to be the complex dilatation of a locally quasiconformal mapping f which can be homeomorphically extended to the closed unit disc.     

Bohr radius for locally univalent harmonic mappings

We consider the class of all sense‐preserving harmonic mappings of the unit disk , where h and g are analytic with , and determine the Bohr radius if any one of the following conditions holds:

• 1. h is bounded in .
• 2. h satisfies the condition in with .
• 3. both h and g are bounded in .
• 4. h is bounded and .

We also consider the problem of determining the Bohr radius when the supremum of the modulus of the dilatation of f in is strictly less than 1. In addition, we determine the Bohr radius for the space of analytic Bloch functions and the space of harmonic Bloch functions. The paper concludes with two conjectures.     

Harmonic mappings and quasiconformal mappings

Work done with support from NSF grant MCS 83-00248.     

Discrete quasiconformal mappings

Summary Some basic properties of quasiconformal mappings are reviewed which leads to the development of discrete analogs of quasiconformal mappings of a multiply connected domain onto a rectangular domain with slits. These discrete mappings can be constructed by solving a system of equations. Comparison of results for a special conformal mapping suggest that the discrete mapping is a good approximation to the quasiconformal mapping.

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Zusammenfassung Es werden einige grundlegende Eigenschaften quasikonformer Abbildungen diskutiert, und es wird ein diskretes Analogon solcher Abbildungen aufgestellt, mit Hilfe dessen mehrfach zusammenhängende Bereiche auf Rechteckbereiche mit Schlitzen abgebildet werden können. Ein Vergleich der Resultate für eine spezielle konforme Abbildung zeigt, dass die diskrete Abbildung eine gute Approximation der quasikonformen Abbildung liefert.

     

Distortion functions for plane quasiconformal mappings

The authors study two well-known distortion functions, λ(K) andϕ _K(r), of the theory of plane quasiconformal mappings and obtain several new inequalities for them. The proofs make use of some properties of elliptic integrals. The work of the first author was supported in part by a grant from the United States National Science Foundation. The work of the first and second authors was supported in part by a grant from the Academy of Finland.     

Maximum principle for spatial quasiconformal mappings

Translated from Matematicheskie Zametki, Vol. 50, No. 6, pp. 14–23, December, 1991.     

On mean quasiconformal mappings

The research was partly supported by the International Science Foundation (Grant U 96 000).     

The invariance of inner and outer linearly locally connected sets under quasiconformal mappings

In this paper, we prove that the inner and outer linearly locally connected sets of Rn are invariant respectively under quasiconformal mappings which fixed the point at infinity. On the other hand, we prove the converses of above results to be true also.     

Extremal polygonal quasiconformal mappings and biLipschitz mappings

We show that the extremal polygonal quasiconformal mappings are biLipschitz with respect to the hyperbolic metric in the unit disk.     

Unique extremality of quasiconformal mappings

In this paper, we study the conditions under which unique extremality of quasiconformal mappings occurs and provide a broader point of view of this phenomenon. Additional information is obtained by means of specialized constructions. In particular, we generalize the construction theorem in [BLMM], thus providing a more basic understanding of it. We also generalize the notion of unique extremality and give an analytic characterization of the generalized concept. Research partially supported by MNTRS, Serbia, Grant No. 144 020