ACTA MATHEMATICA VIETNAMICA 113 Volume 30, Number 2, 2005, pp. 113-121 DIFFERENTIAL SUBORDINATION ASSOCIATED WITH LINEAR OPERATORS DEFINED FOR MULTIVALENT FUNCTIONS V. RAVICHANDRAN, M. DARUS, M. HUSSAIN KHAN AND K. G. SUBRAMANIAN Abstract. In this paper we give certain sufficient conditions for functions defined through the Dziok-Srivastava linear operator and the multiplier trans- formation. 1. Introduction Let A p denote the class of all analytic functions f (z) of the form f (z)= z p + ∞  k=p+1 a k z k (1.1) (z ∈ ∆ := {z : z ∈ C and |z| < 1}; p<k; p, k ∈ N := {1, 2, 3, ···}) and A := A 1 . Recently several authors [8, 12, 13, 16, 17, 18, 19, 25] obtained sufficient conditions associated with starlikeness in terms of the expression zf ′ (z) f (z) + α z 2 f ′′ (z) f (z) . In fact, Ravichandran [19] obtained the following more general result: Theorem 1.1. [19, Theorem 3, p.44] Let q(z) be convex univalent and 0 <α ≤ 1, Re  1 - α α +2q(z)+  1+ zq ′′ (z) q ′ (z)  > 0. If f ∈A satisfies zf ′ (z) f (z) + α z 2 f ′′ (z) f (z) ≺ (1 - α)q(z)+ αq 2 (z)+ αzq ′ (z), then zf ′ (z) f (z) ≺ q(z) and q(z) is the best dominant. Also the following extension of a result of Darus and Frasin [6] was obtained: Received June 26, 2004. 2000 Mathematics Subject Classification. 30C45, 30C80. Key words and phrases. Analytic functions, starlike functions, Dziok-Srivastava linear oper- ator, multiplier transformation, Hadamard product (or convolution).