On the length–area dilatation

Abstract

Beginning in 1956, in order to obtain type criteria by quasiconformal (and more general) mappings f: G → G′, where G and G′ are domains in the plane or on Riemann surfaces, we considered at the regular points of f a length–area dilatation d depending not only on the dilatation quotient but also on the orientation of the characteristic ellipses [Andreian Cazacu, C., 1960, Sur les relations entre les fonctions caractéristiques de la pseudo-analyticité. In: Lucrările celui de al IV-lea Congres al Matematicienilor Români, Bucureşti 1956 (Bucureşti: Editura Academiei Republicii Populare Române), pp. 82–85; Andreian Cazacu, C., 1957, Sur les transformations pseudo-analytiques. Revue de Mathématiques Pures et Appliquées, 2, 383–397; Andreian Cazacu, C., 1964, Sur l'application de la longueur extrémale dans la théorie des représentations quasi-conformes. Mathematica (Cluj), 6(29), 1, 5–10; Andreian Cazacu, C., 1965, Sur un problème de L.I. Volkovyski. Revue Roumaine de Mathématiques Pures et Appliquées, 10, 1, 43–63; Andreian Cazacu, C., 1971, Influence of the orientation of the characteristic ellipses on the properties of the quasiconformal mappings. In: Proceedings of the Romanian-Finnish Seminar on Teichmüller Spaces and Quasiconformal Mappings. Braşov 1969 (Bucharest: Publishing House of the Academy of the Socialist Republic of Romania), pp. 65–85; Andreian Cazacu, C., 1974, A generalization of the quasiconformality. In: Topics in Analysis, Colloquium on Mathematical Analysis, Jyväskylä 1970, Lecture Notes in Mathematics, Vol. 419 (Berlin-Heidelberg-New York: Springer-Verlag), pp. 4–17].

Last time the length–area dilatation found (under diverse names and notations) significant applications [Marković, V. and Mateljević, M., 1997, New versions of Grötzsch principle and Reich–Strebel inequality. Matematichki Vesnik, 49, 235–239; Marković, V. and Mateljević, M., 1999, New versions of the main inequality and uniqueness of harmonic maps. Journal d'Analyse Mathématique, 79, 315–334; Gutlyanskij, V. and Martio, O., 2003, Conformality of a quasiconformal mapping at a point. Journal d'Analyse Mathématique, 91, 179–192] and the aim of this work is to point out its role in these researches, to establish connexions between old and new results. To simplify, we deal with domains in the plane since the essential is already evident in this frame.

Keywords:

• Quasiconformal mappings
• Length–area method
• Grötzsch principle
• Extremal length
• Module inequalities
• AMS 2000 Mathematics Subject Classification: 30C62

Notes

Dedicated to Professor Heinrich Begehr on the occasion of his 65th birthday.

Email: [email protected]

Email: [email protected]

Additional information

Notes on contributors

Cabiria Andreian Cazacu *

Dedicated to Professor Heinrich Begehr on the occasion of his 65th birthday. Email: [email protected]