Abstract. In this present investigation, authors introduce certain subclasses of starlike and convex functions of complex order b, using a linear multiplier differential operator D m λ,µ f (z). In this paper, for these classes the Fekete-Szegö problem is completely solved. Various new special cases of our results are also pointed out.
IntroductionLet A denote the family of functions f of the forma n z n which are analytic in the open unit disk U = {z : |z| < 1}. Further, let S denote the class of functions which are univalent in U . It is well-known that for f ∈ S, a 3 − a 2 2 1. A classical theorem of Fekete-Szegö (see [7]) states that for f ∈ S given by (1.1)This inequality is sharp in the sense that for each η there exists a function in S such that equality holds. Later, Pfluger (see [17]) has considered the complex values of η and provided Up to this time, several authors have attempted to extend the above inequality to more general classes of analytic functions. Given 0 α < 1, a function f ∈ A is said to be in the class S * (α) of starlike functions of order α in U ifOn the other hand, a function f ∈ A is said to be in the class of convex functions of order α in U , denoted by C(α), ifA notions of α-starlikeness and α-convexity were generalized onto a complex order α by Nasr and Aouf (see [13]), Wiatrowski (see [21]), Nasr and Aouf (see [14]). In particular, the classes S * = S * (0) and C = C(0) are the familiar classes of starlike and convex functions in U , respectively. The linear multiplier differential operator D m,α λ,µ f was defined by the authors in (see [6]) as followsIf f is given by (1.1) then from the definition of the operator D m λ,µ f (z) it is easy to see thatIt should be remarked that the D m,α λ,µ is a generalization of many other linear operators considered earlier. In particular, for f ∈ A we have the following:the operator investigated by Sălăgean (see [20]).the operator studied by Al-Oboudi (see [2]).• D m λ,µ f (z) the operator firstly considered for 0 µ λ 1, by Răducanu and Orhan (see [19] By giving specific values to the parameters m, b, λ and µ, we obtain the following important subclasses studied by various authors in earlier works, for instance, S m (1 − α, 1, 0) = S m (α) (Sălăgean (see [20])), S 0 (b, 1, 0) =