Das [3] introduced the class of absolute kth-power conservative matrices for k ≥ 1, denoted by B (A k ) . In the present paper, we generalize the class B (A k ) to a general one named B α n , β n ; γ n , δ n ; ϕ and give some sufficient conditions for a matrix belongs to the new class B α n , β n ; γ n , δ n ; ϕ when ϕ is convex. As applications of the general result, we investigate the conservatives of Cesáro matrices and Riesz matrices.