Abstract:
Harmonic and quasiregular mappings are the generalization of conformal mappings and analytic functions in the plane. It is natural to extend the theories of analytic functions and conformal mappings to harmonic and quasiregular cases. In this thesis, we consider questions of their geometry properties and boundary behavior.
For the first part, we study the univalency of linear combinations of two harmonic quasiregular mappings with fixed analytic functions which are $M$-linearly connected. In addition, we consider the relation of bi-Lipschitz of the linear combinations and the fixed function.
For the second part, we generalize the Lipschiz property of the analytic Bloch function under the pseudo-hyperbolic metric to consider the harmonic Bloch and harmonic Bloch-type cases.
For the third part, we consider the boundary Schwarz lemma for harmonic mappings having zero of order $p$.
For the fourth part, we investigate Lindel\"{o}f and Koebe type boundary behavior results for bounded quasiregular mappings in $n$-dimensional Euclidean spaces. Furthermore, the existence of non-tangential limits at a boundary point is considered.
For the last part, we use different methods to study the uniqueness of harmonic mappings obtained by the construction of Sheil-Small, and give the proof of a special case.
Main Advisor | Associate Professor Anti Hermanni Rasila |
Associate Advisor | Professor Emeritus Bshouty Daoud |