Criteria for univalency and quasiconformal extension for harmonic mappings

In this paper, we study quasiconformal extensions of harmonic mappings. Utilizing a complex parameter, we build a bridge between the quasiconformal extension theorem for locally analytic functions given by Ahlfors [‘Sufficient conditions for quasiconformal extension’, Ann. of Math. Stud.79 (1974), 23–29] and the one for harmonic mappings recently given by Hernández and Martín [‘Quasiconformal extension of harmonic mappings in the plane’, Ann. Acad. Sci. Fenn. Math.38 (2) (2013), 617–630]. We also give a quasiconformal extension of a harmonic Teichmüller mapping, whose maximal dilatation estimate is asymptotically sharp.

Download Full-text

Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains

Annales Fennici Mathematici ◽

10.5186/aasfm.2021.4669 ◽

2021 ◽

Vol 46 (2) ◽

pp. 1123-1134

Author(s):

Iason Efraimidis

Keyword(s):

Quasiconformal Extension ◽

Harmonic Mappings ◽

Planar Domains

Download Full-text

A Class of Harmonic Starlike Functions defined by Multiplier Transformation

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.36 ◽

2020 ◽

pp. 455-469

Author(s):

Deepali Khurana ◽

Raj Kumar ◽

Sibel Yalcin

Keyword(s):

Convex Combination ◽

Univalent Functions ◽

Extreme Points ◽

Starlike Functions ◽

Harmonic Mappings ◽

Distortion Bounds ◽

Radius Of Convexity ◽

Univalent Harmonic Mappings ◽

Harmonic Starlike ◽

Multiplier Transformation

We define two new subclasses, $HS(k, \lambda, b, \alpha)$ and \linebreak $\overline{HS}(k, \lambda, b, \alpha)$, of univalent harmonic mappings using multiplier transformation. We obtain a sufficient condition for harmonic univalent functions to be in $HS(k,\lambda,b,\alpha)$ and we prove that this condition is also necessary for the functions in the class $\overline{HS} (k,\lambda,b,\alpha)$. We also obtain extreme points, distortion bounds, convex combination, radius of convexity and Bernandi-Libera-Livingston integral for the functions in the class $\overline{HS}(k,\lambda,b,\alpha)$.

Download Full-text

On a Subclass of Univalent Harmonic Mappings Convex in the Imaginary Direction

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.35 ◽

2020 ◽

pp. 445-454

Author(s):

Deepali Khurana ◽

Sushma Gupta ◽

Sukhjit Singh

Keyword(s):

Present Article ◽

Unit Disk ◽

Imaginary Axis ◽

Harmonic Mappings ◽

Open Unit ◽

Open Unit Disk ◽

Distortion Bounds ◽

Univalent Harmonic Mappings

In the present article, we consider a class of univalent harmonic mappings, $\mathcal{C}_{T} = \left\{ T_{c}[f] =\frac{f+czf'}{1+c}+\overline{\frac{f-czf'}{1+c}}; \; c>0\;\right\}$ and $f$ is convex univalent in $\mathbb{D}$, whose functions map the open unit disk $\mathbb{D}$ onto a domain convex in the direction of the imaginary axis. We estimate coefficient, growth and distortion bounds for the functions of the same class.

Download Full-text