Some Families of Starlike Functions with Negative Coefficients

Abstract

We introduce the subclassP(j,λ,α,n) of starlike functions with negative coefficients by using the differential operatorDnwhich was considered by G. Şt. Sălăgean (“Complex Analysis Fifth Romanian-Finnish Seminar,” pp. 362–372, Lecture Notes in Math., Vol. 1013, Springer-Verlag, Berlin/New York, 1983). Coefficient inequalities, distortion theorems, closure theorems, and some properties involving the modified Hadamard products of several functions belonging to the classP(j,λ,α,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 classP(j,λ,α,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).