The authors extend the summability result of H. Bor and H. S. Özarslan [Rocky Mt. J. Math. 38, No. 3, 801–807 (2008; Zbl 1170.40003)] to a two-parameter quasi-\(f^{\beta,\sigma}\)-power increasing sequence, instead of a quasi-\(\beta\)-power increasing sequence. Several known results are shown to be special cases and consequences of the proved general summability theorem.