ON SOME CLASSES OF ANALYTIC FUNCTIONS DEFINED BY A MULTIPLIER TRANSFORMATION

Abstract. We introduce two new classes of analytic functions defined by applying a multiplier transformation to functions f ∈ A (p) and study some containment properties of this classes. 1. Preliminaries LetH = H (U) denote the class of functions analytic in U = {z ∈ C: |z | < 1}. For n a positive integer and a ∈ C, let H [a, n] = {f ∈ H: f (z) = a+ anzn +...}. For p ∈ N∗, we consider A (p) to be the class of functions of the form f (z) = zp + n=p+1 anz n, which are analytic in the unit disk U. We denote by Q the set of functions f that are analytic and injective on U \ E (f), where E (f) = ζ ∈ ∂U: lim z→ζ f (z) =∞ and are such that f ′ (ζ) 6 = 0 for ζ ∈ ∂U \ E (f). Since we use the terms of subordination and superordination, we review here those definitions. Let f, F ∈ H. The function f is said to be subordinate to F, or F is said to be superordinate to f, if there exists a function w analytic in U, with Received by the editors: 19.04.2007. 2000 Mathematics Subject Classification. 30C80, 30C45. Key words and phrases. multiplier transformation, differential subordinations and superordinations, classes of p-valent functions. Partially supported by UEFISCSU Grant PN-II-ID 524/2007