Moduli of Continuity of Quasiregular Mappings

This thesis consists of Chapters 1 and 2. The main results are contained in the two preprints and two published papers, listed below. Chapter 1 deals with conformal invariants in the euclidean space Rn; n >= 2; and their interrelation. In particular, conformally invariant metrics and balls of the respective metric spaces are studied. Another theme in Chapter 1 is the study of quasiconformal maps with identity boundary values in two diferent cases, the unit ball and the whole space minus two points. These results are based on the two preprints: R. Klen, V. Manojlovic and M. Vuorinen: Distortion of two point normalized quasiconformal mappings, arXiv:0808.1219[math.CV], 13 pp., V. Manojlovic and M. Vuorinen: On quasiconformal maps with identity boundary values, arXiv:0807.4418[math.CV], 16 pp. Chapter 2 deals with harmonic quasiregular maps. Topics studied are: Preservation of modulus of continuity, in particular Lipschitz continuity, from the boundary to the interior of domain in case of harmonic quasiregular maps and quasiisometry property of harmonic quasiconformal maps. Chapter 2 is based mainly on the two published papers: M. Arsenovic, V. Kojic and M. Mateljevic: On Lipschitz continuity of harmonic quasiregular maps on the unit ball in Rn., Ann. Acad. Sci. Fenn. Math. 33 (2008), no. 1, 315-318. V. Kojic and M. Pavlovic: Subharmonicity of jfjp for quasiregular harmonic functions, with applications, J. Math. Anal. Appl. 342 (2008) 742-746