On Some Subclass of Harmonic Close-to-convex Mappings

TL;DR

This paper investigates a subclass of harmonic functions in the unit disk, proving a coefficient conjecture, establishing growth and convolution properties, and analyzing partial sums for close-to-convexity.

Contribution

It proves the coefficient conjecture of Clunie and Sheil-Small for the class , and explores growth, convolution, and partial sum properties of these harmonic mappings.

Findings

Proved the coefficient conjecture for  class.

Established growth and convolution properties.

Determined radius for close-to-convexity of partial sums.

Abstract

Let $\mathcal{H}$ denote the class of harmonic functions $f$ in $\mathbb{D}:= \{z\in \mathbb{C}:|z| < 1\}$ normalized by $f(0) = 0 = f_z(0) -1$. For $\alpha \geq 0$, we consider the following class $$\mathcal{W}^0_{\mathcal{H}}(\alpha):= \{f = h + \overline{g}\in\mathcal{H}: {\rm Re\,}(h'(z) + \alpha z h''(z)) >|g'(z) + \alpha z g''(z)|, \quad z\in \mathbb{D}\}. $$ In this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class $\mathcal{W}^0_{\mathcal{H}}(\alpha)$. We also prove growth theorem, convolution, convex combination properties for functions in the class $\mathcal{W}^0_{\mathcal{H}}(\alpha)$. Finally, we determine the value of $r$ so that the partial sums of functions in the class $\mathcal{W}^0_{\mathcal{H}}(\alpha)$ are close-to-convex in $|z|<r$.