Analele Universit˘ at ¸ii Oradea Fasc. Matematica, Tom XXI (2014), Issue No. 1, 183–190 SANDWICH THEOREMS FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS DEFINEND BY CONVOLUTION STRUCTURE WITH GENERALIZAD OPERATOR ABBAS KAREEM WANAS 1 AND AHMED SALLAL JOUDAH 2 Abstract. The purpose of the present paper is to derive sandwich results involving Hadamard product for certain normalized analytic functions with generalized operator in the open unit disk. 1. Introduction Let H be the class of analytic functions in the open unit disk U = {z ∈ C : |z| < 1} .For n a positive integer and a ∈ C, let H [a,n] be the subclass of H consisting of functions of the form f (z)= a + a n z n + a n+1 z n+1 + ··· (a ∈ C) . (1.1) Also, let A be the subclass of H consisting of functions of the form: f (z)= z + ∞  n=2 a n z n . (1.2) Let f,g ∈ H. The function f is said to be subordinate to g , or g is said to be superordinate to f , if there exists a schwarz function w analytic in U with w(0) = 0 and |w(z)| < 1(z ∈ U ) such that f (z)= g(w(z)). In such a case we write f ≺ g or f (z) ≺ g(z)(z ∈ U ). If g is univalent in U , then f ≺ g if and only if f (0) = g(0) and f (U ) ⊂ g(U ). Let p,h ∈ H and ψ(r,s,t; z): C 3 × U → C. If p and ψ(p(z),zp ′ (z),z 2 p ′′ (z); z) are univalent functions in U and if p satisfies the second -order differential superordination h(z) ≺ ψ(p(z),zp ′ (z),z 2 p ′′ (z); z), (1.3) then p is called a solution of the differential superordination (1.3). (If f is subordinate to g , then g is superordinate to f ). An analytic function q is called a subordinate of (1.3), if q ≺ p for all the functions p satisfying (1.3). An univalent subordinat q that satisfies q ≺ q for all the subordinants q of (1.3) is called the best subordinant. Recently Miller and Mocanu [10] obtained conditions on the functions h,q and ψ for which the following implication holds: h(z) ≺ ψ(p(z),zp ′ (z),z 2 p ′′ (z); z) ⇒ q(z) ≺ p(z). For the functions f ∈ A, f (z)= z + ∑ ∞ n=2 a n z n and g ∈ A defined by g(z)= z + ∑ ∞ n=2 b n z n , we define the Hadamard product (or convolution ) of f and g by (f ∗ g)(z)= z + ∑ ∞ n=2 a n b n z n =(g ∗ f )(z). 2000 Mathematics Subject Classification. 30C45, 30C80. Key words and phrases. Analytic functions, Differential subordination, Differential Superordination, Hadamard product, Dominant, Subordinant, Integral operator. 183