Some Families of Starlike Functions with Negative Coefficients

We introduce the subclass P j, l, a, n. of starlike functions with negative coefficients by using the differential operator Dn which was considered by G. S¸t. S˘al˘agean ‘‘Complex Analysis Fifth Romanian-Finnish Seminar,’’ pp. 362]372, Lecture Notes in Math., Vol. 1013, Springer-Verlag, BerlinrNew York, 1983.. Coefficient inequalities, distortion theorems, closure theorems, and some properties involving the modified Hadamard products of several functions belonging to the class P j, l, a, n.are obtained. We also determine the radii of close-to-convexity, starlikeness, and convexity for, and consider integral operators associated with, functions belonging to the class P j, l, a, n.. Finally, we extend some of the aforementioned distortion theorems to hold true for certain operators of fractional calculus that is, fractional integral and fractional derivative..