📝 SummarXiv

Abstract

Let $\mathcal{H}$ be the class of harmonic functions $f=h+\overline{g}$ in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$, where $h$ and $g$ are analytic in $\mathbb{D}$ with the normalization $h(0)=g(0)=h'(0)-1=0$. Let $\mathcal{P}_{\mathcal{H}}^0(\alpha,M)$ denote the subclass of $\mathcal{H}$ in $\mathbb{D}$ satisfying $\text{Re}\left((1-\alpha)h'(z)+\alpha zh''(z)\right)>-M+\left|(1-\alpha)g'(z)+\alpha zg''(z)\right|$ with $g'(0)=0$, $M>0$ and $\alpha\in(0,1]$. In this paper, we investigate some fundamental properties of functions belonging to the class $\mathcal{P}_{\mathcal{H}}^0(\alpha,M)$, including coefficient bounds, growth estimates, convexity, starlikeness, convex combinations, and convolution.