An Application of Jackson's $(p, q)$-Derivative to a Subclass of Starlike Functions with Negative Coefficients
In this paper, we introduce and investigate the subclass $\mathcal{P}_{p,q}^{ξ,κ}(τ, η)$ of starlike functions with negative coefficients by using the differential operator $Υ_{τ,p,q}^{ξ,κ}$. Coefficient inequalities, growth and distortion theorems, closure theorems, and some properties of several functions belonging to this class are obtained. We also determine the radii of close-to-convexity, starlikeness, and convexity for functions belonging to the class $\mathcal{P}_{p,q}^{ξ,κ}(τ, η)$. Furthermore, we obtain the integral means inequality and neighborhood results for functions belonging to the class $\mathcal{P}_{p,q}^{ξ,κ}(τ, η)$. The results presented in this paper generalize or improve those in related works of several earlier authors.