Some families of starlike functions with negative coefficients (Q1125849)

The authors introduce the subclass \({\mathcal P} (j, \lambda, \alpha, n)\) of starlike functions with negative coefficients by using the differential operator \({\mathcal D}^n\) which was considered by \textit{G. Şt. Sălăgean} [Lect. Notes Math., 1013, 362-372 (1983; Zbl 0531.30009)]. Coefficient inequalities, distortion theorems, closure theorems, and some properties involving the modified Hadamard products of several functions belonging to the class \({\mathcal P} (j, \lambda, \alpha, n)\) are obtained. They also determine the radii of close-to-convexity, starlikeness, and convexity for, and consider integral operators associated with, functions belonging to the class \({\mathcal P} (j, \lambda, \alpha,n)\). Finally, they extend some of the aforementioned distortion theorems to hold true for certain operators of fractional calculus (that is, fractional integral and fractional derivative).