Das [3] introduced the class of absolute $k$th-power conservative matrices for $k\geq1$, denoted by $B(A_k)$. In the present paper, we generalize the class $B(A_k)$ to a general one named $B(\alpha_n,\beta_n;\gamma_n,\delta_n;\varphi)$ and give some sufficient conditions for a matrix belongs to the new class $B(\alpha_n,\beta_n;\gamma_n,\delta_n;\varphi)$ when $\varphi$ is convex. As applications of the general result, we investigate the conservatives of Cesáro matrices and Riesz matrices.