On Convolution and Convex Combination of Harmonic Mappings

In this paper, the subclass of harmonic univalent functions by shearing construction is studied and this subclass of harmonic mappings needs a necessary and adequate condition to be convex in the horizontal direction. Furthermore, convolutions of two special subclasses of univalent harmonic mappings are shown to be convex in the horizontal direction. Also, the family of univalent harmonic mappings of the unit disk onto a region convex in the direction of the imaginary axis is introduced. Sufficient conditions for convex combinations of harmonic mappings of this family to be univalently convex in the direction of the imaginary axis are obtained.

A Class of Harmonic Starlike Functions defined by 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 $\overline{HS}(k,\lambda,b,\alpha)$.

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

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.