Certain New Class of Harmonic Functions Involving Quantum Calculus

Abstract

Here in this paper, we are using the concepts of $q$ -calculus operator theory associated with harmonic functions and define the $q$ -Noor integral operator for harmonic functions $f\in {\mathcal{H}}^0.$ We investigate a new class ${\mathcal{S}}_{\mathcal{H}}^0\left(m,q,\alpha \right)$ of harmonic functions $f\in {\mathcal{H}}^0$ . In this class, we prove a necessary and sufficient convolution condition for the functions $f\in {\mathcal{H}}^0$ and also we proved that this sufficient coefficient condition is sense preserving and univalent in the class ${\mathcal{S}}_{\mathcal{H}}^0\left(m,q,\alpha \right)$ . It is proved that this coefficient condition is necessary for the functions in its subclass $\mathcal{T}{\mathcal{S}}_{\mathcal{H}}^0\left(m,q,\alpha \right)$ . By using this necessary and sufficient coefficient condition, we obtained results based on the convexity and compactness and results on the radii of $q$ -starlikeness and $q$ -convexity of order $\alpha$ in the class $\mathcal{T}{\mathcal{S}}_{\mathcal{H}}^0\left(m,q,\alpha \right)$ . Also we obtained extreme points for the functions in the class $\mathcal{T}{\mathcal{S}}_{\mathcal{H}}^0\left(m,q,\alpha \right).$