Harmonic shears of slit and polygonal mappings
نویسندگان
چکیده
Aalto University, P.O. Box 11000, FI-00076 Aalto www.aalto.fi Author Saminathan Ponnusamy, Tri Quach, Antti Rasila Name of the publication Harmonic shears of slit and polygonal mappings Publisher School of Science Unit Department of Mathematics and Systems Analysis Series Aalto University publication series SCIENCE + TECHNOLOGY 3/2012 Field of research Harmonic Shears of Slit and Polygonal Mappings Saminathan Ponnusamy, Tri QuachR, and Antti Rasila Abstract. In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations ω. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using MATHEMATICA.
منابع مشابه
Convex combinations of harmonic shears of slit mappings
In this paper, we study the convex combinations of harmonic mappings obtained by shearing a class of slit conformal mappings. Sufficient conditions for the convex combinations of harmonic mappings of this family to be univalent and convex in the horizontal direction are derived. Several examples of univalent harmonic mappings constructed by using these methods are presented to illustrate...
متن کاملSOME HARMONIC n - SLIT MAPPINGS
The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, , such that f = h+g where h(z) = z+ P 1 2 akz , g(z) = P 1 1 bkz . S H will denote the subclass with b1 = 0. We present a collection of n-slit mappings (n 2) and prove that the 2-slit mappings are in SH while for n 3 the mappings are in S H . Finally we show that these mappings establish the sharpne...
متن کاملOn the Linear Combinations of Slanted Half-Plane Harmonic Mappings
In this paper, the sufficient conditions for the linear combinations of slanted half-plane harmonic mappings to be univalent and convex in the direction of $(-gamma) $ are studied. Our result improves some recent works. Furthermore, a illustrative example and imagine domains of the linear combinations satisfying the desired conditions are enumerated.
متن کاملThe starlikeness, convexity, covering theorem and extreme points of p-harmonic mappings
The main aim of this paper is to introduce three classes $H^0_{p,q}$, $H^1_{p,q}$ and $TH^*_p$ of $p$-harmonic mappings and discuss the properties of mappings in these classes. First, we discuss the starlikeness and convexity of mappings in $H^0_{p,q}$ and $H^1_{p,q}$. Then establish the covering theorem for mappings in $H^1_{p,q}$. Finally, we determine the extreme points of the class $TH^*_{p}$.
متن کاملThe Inner Mapping Radius of Harmonic Mappings of the Unit Disk
The class SH consists of univalent, harmonic, and sense-preserving functions f in the unit disk, ∆, such that f = h + g where h(z) = z + ∑∞ 2 akz , g(z) = ∑∞ 1 bkz . Using a technique from Clunie and Sheil-Small, we construct a family of 1-slit mappings in SH by varying ω(z) = g ′(z)/f ′(z). As ω(z) changes, the tip of the slit slides along the negative real axis from the point 0 to −1. In doin...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Applied Mathematics and Computation
دوره 233 شماره
صفحات -
تاریخ انتشار 2014