Inclusion Properties for Certain Subclasses of Analytic Functions Defined by a Linear Operator

which are analytic in the open unit disk U {z ∈ C : |z| < 1}. If f and g are analytic in U, we say that f is subordinate to g, written f ≺ g or f z ≺ g z if there exists an analytic function w in U with w 0 0 and |w z | < 1 for z ∈ U such that f z g w z . We denote by S∗, K, and C the subclasses of A consisting of all analytic functions which are, respectively, starlike, convex, and close-to-convex in U. LetN be the class of all functions φ which are analytic and univalent in U and for which φ U is convex with φ 0 1 and Re{φ z } > 0 for z ∈ U. Making use of the principle of subordination between analytic functions, many authors investigated the subclasses S∗ φ , K φ , and C φ, ψ of the class A for φ, ψ ∈ N cf. 1, 2 , which are defined by