doi: 10.2478/v10062-010-0009-7 ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN – POLONIA VOL. LXIV, NO. 2, 2010 SECTIO A 1–14 MOHAMED K. AOUF and TAMER M. SEOUDY On differential sandwich theorems of analytic functions defined by certain linear operator Abstract. In this paper, we obtain some applications of first order differ- ential subordination and superordination results involving certain linear op- erator and other linear operators for certain normalized analytic functions. Some of our results improve and generalize previously known results. 1. Introduction. Let H (U ) be the class of analytic functions in the open unit disk U = {z ∈ C : |z | < 1} and let H [a, k] be the subclass of H (U ) consisting of functions of the form: (1.1) f (z )= a + a k z k + a k+1 z k+1 ... (a ∈ C ). For simplicity H [a]= H [a, 1]. Also, let A be the subclass of H (U ) consisting of functions of the form: (1.2) f (z )= z + ∞  k=2 a k z k . If f , g ∈ H (U ), we say that f is subordinate to g or f is superordinate to g, written f (z ) ≺ g(z ) if there exists a Schwarz function ω, which (by definition) is analytic in U with ω(0) = 0 and |ω(z )| < 1 for all z ∈ U, such 2000 Mathematics Subject Classification. 30C45. Key words and phrases. Analytic function, Hadamard product, differential subordina- tion, superordination, linear operator.