For given analytic functions ϕ ( z ) = z + ∑ m = 2 ∞   λ m   z m , ψ ( z ) = z + ∑ m = 2 ∞   μ m   z m in U = { z | | z | < 1 } with λ m ≥ 0 , μ m ≥ 0 and λ m ≥ μ m , let E n ( ϕ , ψ ; A , B ) be the class of analytic functions f ( z ) = z + ∑ m = 2 ∞ a m   z m in U such that ( f * Ψ ) ( z ) ≠ 0 and D n + 1 ( f * ϕ ) ( z ) D n ( f * Ψ ) ( z ) ≪ 1 + A z 1 + B z ,             − 1 ≤ A < B ≤ 1 ,     z ∈ U , where D n h ( z ) = z ( z n − 1 h ( z ) ) ( n ) / n ! ,       n ∈ N 0 = { 0 , 1 , 2 , … } is the n th Ruscheweyh derivative; ≪ and * denote subordination and the Hadamard product, respectively. Let T be the class of analytic functions in U of the form f ( z ) = z − ∑ m = 2 ∞ a m   z m ,     a m ≥ 0 , and let E n [ ϕ , ψ ; A , B ] = E n ( ϕ , ψ ; A , B ) ∩ T . Coefficient estimates, extreme points, distortion theorems and radius of starlikeness and convexity are determined for functions in the class E n [ ϕ , ψ ; A , B ] . We also consider the quasi-Hadamard product of functions in E n [ z / ( 1 − z ) , z / ( 1 − z ) ; A , B ] and E n [ z / ( 1 − z ) 2 , z / ( 1 − z ) 2 ; A , B ] .