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Introduction

In: Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics

Author

Listed:
  • Vesna Todorčević

    (University of Belgrade, Faculty of Organizational Sciences
    Serbian Academy of Sciences and Arts, Mathematical Institute)

Abstract

Geometric Function Theory began as a branch of Complex Analysis dealing with geometric aspects of analytic functions, but has since grown considerably, both in scope and in methodology. It considers, for example, the class of quasiregular mappings proven to be a natural and especially fruitful generalization of analytic functions in the planar case. Another class considered is the class of quasiconformal mappings characterized by the property that there is a constant C ≥ 1 such that infinitesimal spheres are mapped onto infinitesimal ellipsoids in such a manner that the ratio of the longest axis to the shortest axis is bounded from above by C. Injective quasiregular mappings are quasiconformal and conformal mappings in the plane are both harmonic and quasiconformal. Moreover, harmonic mappings are smooth and if they are also quasiregular they are locally quasiconformal in higher dimensions. This gives us a motivation to study harmonic quasiconformal mappings in higher dimensions. Today the study of these classes of mappings is recognized as an important research area of Geometric Function Theory.

Suggested Citation

  • Vesna Todorčević, 2019. "Introduction," Springer Books, in: Harmonic Quasiconformal Mappings and Hyperbolic Type Metrics, chapter 0, pages 1-4, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-22591-9_1
    DOI: 10.1007/978-3-030-22591-9_1
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