Let K denote the class of functions g ( z ) = z + a 2 z 2 + ⋯ which are regular and univalently convex in the unit disc E . In the present note, we prove that if f is regular in E , f ( 0 ) = 0 , then for g ∈ K , f ( z ) + α z f ′ ( z )   ≺   g ( z ) + α z g ′ ( z ) in E implies that f ( z ) ≺ g ( z ) in E , where 0$"> α > 0 is a real number and the symbol “ ≺ ” stands for subordination.