On Properties of $q$-Close-to-Convex Harmonic Functions of Order $\alpha$

Abstract

In this paper, a novel subclass, denoted as $\mathcal{PH}(q, \alpha)$, is unveiled within the domain of harmonic functions in the open unit disk $\mathbb{E}$. This subclass, comprised of functions $\mathfrak{f}=\mathfrak{u}+\overline{\mathfrak{v}}\in \mathcal{SH}^{0}$, is characterized by a specific inequality involving the $q$-derivative operator. Through meticulous analysis, it is demonstrated that functions belonging to $\mathcal{PH}(q, \alpha)$ exhibit remarkable close-to-convexity properties. Furthermore, diverse results such as distortion theorem, coefficient bounds, and a sufficient coefficient condition are yielded by the exploration. Additionally, the closure properties of $\mathcal{PH}(q, \alpha)$ under convolution operations and convex combination are elucidated, underscoring its structural coherence and relevance in the broader context of harmonic mappings.

Keywords

• q-Derivative
• harmonic functions
• q-close-to-convex functions
• coefficient bounds
• distortion.

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