Geometric properties of harmonic and polyharmonic mappings

Thumbnail Image
Files
isbn9789526402956.pdf (289.8 KB)
Journal Title
Journal ISSN
Volume Title
School of Science | Doctoral thesis (article-based) | Defence date: 2021-04-14
Unless otherwise stated, all rights belong to the author. You may download, display and print this publication for Your own personal use. Commercial use is prohibited.
Author
Date
2021
Department
Matematiikan ja systeemianalyysin laitos
Department of Mathematics and Systems Analysis
Major/Subject
Mcode
Degree programme
Language
en
Pages
28 + app. 64
Series
Aalto University publication series DOCTORAL DISSERTATIONS, 31/2021
Abstract
This dissertation lies in the areas of Complex Analysis and Geometric Function Theory. Its subject concerns mainly the classes of harmonic and polyharmonic mappings as well as their relation to analytic functions. Harmonic mappings are defined as the solutions to the complex Laplacian equation. Since every analytic function solves the Laplacian equation, the harmonic mappings have been widely studied as generalisations of the analytic functions. Polyharmonic mappings form the wider class of functions which solve the p-Laplacian equation, for a certain positive p. In the first paper, we improve the harmonic analogue of the classical Bohr inequality, proved by Kayumov, Ponnusamy, and Shakirov. We show that anon-negative term can be added and still obtain the same Bohr radius. In the second paper, we examine how we can improve known results for analytic functions, given by R. Fournier and St. Ruscheweyh, for Bohr's phenomenon in more general domains. We also prove a sharp Bohr's inequality for harmonic mappings defined in disks larger than the unit disk. In the third paper, we study the behaviour of harmonic mappings, defined either in the unit disk or in the upper half-plane, near the boundary.We also find sufficient conditions in order for the boundary function of such a mapping to be continuous, based on a result of Bshouty, Lyzzaik, and Weitsman. In the fourth paper, we show that Radó's theorem remains true for the wider class of polyharmonic mappings. We also obtain results about the continuity of the boundary function of a polyharmonic mapping. At the end, we present a method for constructing a fully close-to-convex biharmonic mapping by using a convex harmonic one.  
Description
Defense is held on 14.4.2021 12:00 – 16:00 Via remote technology (Zoom), https://aalto.zoom.us/j/63792293866
Supervising professor
Kinnunen, Juha, Prof., Aalto University, Department of Mathematics and Systems Analysis, Finland
Thesis advisor
Rasila, Antti, Prof., GTIIT, China
Keywords
complex analysis, harmonic mappings, polyharmonic mappings, Bohr's inequality, boundary behavior, Radó’s theorem
Other note
Parts
  • [Publication 1]: S. Evdoridis, S. Ponnusamy, and A. Rasila. Improved Bohr’s inequality for locally univalent harmonic mappings. Indagationes Mathematicae, 30, 201-213, January 2019.
    Full text in Acris/Aaltodoc: http://urn.fi/URN:NBN:fi:aalto-201901141087
    DOI: 10.1016/j.indag.2018.09.008 View at publisher
  • [Publication 2]: S. Evdoridis, S. Ponnusamy, and A. Rasila. Improved Bohr’s inequality for shifted disks. Results in Mathematics, , 9 January 2021.
    DOI: 10.1007/s00025-020-01325-x View at publisher
  • [Publication 3]: D. Bshouty, J. Chen, S. Evdoridis, and A. Rasila. Koebe and Carathéodory type boundary behavior results for harmonic mappings. Acceptedfor publication in Complex Variables and Elliptic Equations, 16 December 2020.
    DOI: 10.1080/17476933.2020.1851212 View at publisher
  • [Publication 4]: D. Bshouty, S. Evdoridis, and A. Rasila. A note on polyharmonic mappings. Submitted to Computational Methods and Function Theory, 8 December 2020
Citation
Permanent link to this item
https://urn.fi/URN:ISBN:978-952-64-0295-6
Collections
[diss] Perustieteiden korkeakoulu / SCI