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 ${\mathcal{P}}_{p,q}^{\xi ,\kappa }(\tau ,\eta )$ of starlike functions with negative coefficients by using the differential operator $\Upsilon_{\tau ,p,q}^{\xi ,\kappa}$. 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 ${\mathcal{P}}_{p,q}^{\xi ,\kappa }(\tau ,\eta )$. Furthermore, we obtain the integral means inequality and neighborhood results for functions belonging to the class ${\mathcal{P}}_{p,q}^{\xi ,\kappa }(\tau ,\eta )$. The results presented in this paper generalize or improve those in related works of several earlier authors.