Conservative Matrices in Summability of Series

Abstract

Das \cite{d} introduced the class of absolute $k$th-power conserviative matrices for $k\geq1$, denoted by $B\left( A_{k}\right) .$ In the present paper, we generalize the class $B\left( A_{k}\right) $ to a general one named $B\left( \alpha_{n}, \beta_{n}, \gamma_{n},\delta_{n},\varphi\right) $ and give some sufficient conditions for a matrix belongs to the new class $B\left( \alpha_{n}, \beta_{n}; \gamma_{n}, \delta_{n};\varphi\right) $ when
$\varphi$ is convex$.$ As applications of the general result, we investigate the conservativties of Ces\'{a}ro matrices and Riesz matrices.