Shear Construction Of Certain Harmonic Univalent Functions And Weierstrass-Enneper Representation

Geometric function theory is an intriguing field of study because harmonic maps and the minimal surfaces are connected and also it is used in many other fields. The major part of this thesis consists of several original results on harmonic functions and their minimal surface connections. Initially, we present two generalized harmonic univalent functions using the shearing construction, determine a horizontal convexity criterion for these generalized univalent harmonic mappings, and look into the directional convexity and univalency of special subclasses of harmonic mapping convolutions

Convolutions of slanted half-plane harmonic mappings

Let ${\mathcal S^0}(H_{\gamma})$ denote the class of all univalent, harmonic, sense-preserving and normalized mappings $f$ of the unit disk \ID onto the slanted half-plane $H_\gamma :=\{w:\,{\rm Re\,}(e^{i\gamma}w) >-1/2\}$ with an additional condition $f_{\bar{z}}(0)=0$. Functions in this class can be constructed by the shear construction due to Clunie and Sheil-Small which allows by examining their conformal counterpart. Unlike the conformal case, convolution of two univalent harmonic convex mappings in \ID is not necessarily even univalent in \ID. In this paper, we fix $f_0\in{\mathcal S^0}(H_{0})$ and show that the convolutions of $f_0$ and some slanted half-plane harmonic mapping are still convex in a particular direction. The results of the paper enhance the interest among harmonic mappings and, in particular, solves an open problem of Dorff, et. al. \cite{DN} in a more general setting. Finally, we present some basic examples of functions and their corresponding convolution functions with specified dilatations, and illustrate them graphically with the help of MATHEMATICA software. These examples explain the behaviour of the image domains.Comment: 15 pages, preprint of December 2011 (submitted to a journal for publication

Several properties of a class of generalized harmonic mappings

We call the solution of a kind of second order homogeneous partial differential equation as real kernel alpha-harmonic mappings. In this paper, the representation theorem, the Lipschitz continuity, the univalency and the related problems of the real kernel alpha-harmonic mappings are explored

Convolution of some slanted half-plane mappings with harmonic strip mappings

In this paper, we show that the convolution of generalized half-plane mapping and harmonic vertical strip mapping with dilatation eⁱᶱ zⁿ (n ∈ N, θ ∈ R) is convex in a particular direction and also solve the problem proposed by Z. Liu et al. [Convolutions of harmonic half-plane mappings with harmonic vertical strip mappings, Filomat, 31 (2017), no. 7, 1843–1856].Publisher's VersionPMID-123

Note on the convolution of harmonic mappings

Dorff et al. \cite{DN} formulated a question concerning the convolution of two right half-plane mappings, where the normalization of the functions was considered incorrectly. In this paper, we have reformulated the open problem in correct form and provided a solution to it in a more general form. In addition, we also obtain two new theorems which correct and improve some other results.Comment: 11 pages; An extended version of this article was in a couple of conferences, and also in later workshops in Chennai during 2017 in India. This version will appear in Bulletin of the Australian Mathematical Societ