Stud. Univ. Babe¸ s-Bolyai Math. 57(2012), No. 1, 25–41 Some properties of certain class of multivalent analytic functions Mohamed Kamal Aouf, Rabha Mohamed El-Ashwah and Ekram Elsayed Ali Abstract. In this paper we introduce a certain general class Φ β p (a, c, A, B)(β ≥ 0, a> 0, c> 0, -1 ≤ B<A ≤ 1, p ∈ N = {1, 2, ...}) of multivalent analytic functions in the open unit disc U = {z : |z| < 1} involving the linear operator Lp(a, c). The aim of the present paper is to investigate various properties and characteristics of this class by using the techniques of Briot-Bouquet differential subordination. Also we obtain coefficient estimates and maximization theorem concerning the coefficients. Mathematics Subject Classification (2010): 30C45. Keywords: Analytic, multivalent, differential subordination. 1. Introduction Let A(p) denote the class of functions of the form: f (z)= z p + ∞  k=1 a p+k z p+k (p ∈ N = {1, 2, ....}), (1.1) which are analytic and p-valent in the open unit disc U = {z : |z| < 1}. Let Ω denotes the class of bounded analytic functions w(0) = 0 and |w(z)|≤|z| for z ∈ U . If f and g are analytic in U , we say that f subordinate to g, written symbolically as follows: f ≺ g (z ∈ U ) or f (z) ≺ g(z), if there exists a Schwarz function w, which (by definition) is analytic in U with w(0) = 0 and |w(z)| < 1(z ∈ U ) such that f (z)= g(w(z)) (z ∈ U ). In particular, if the function g(z) is univalent in U , then we have the following equivalence (cf., e.g., [5], [18]; see also [19, p. 4]): f (z) ≺ g(z) ⇔ f (0) = g(0) and f (U ) ⊂ g(U ).