The main object of this paper is to prove two general theorems by using a two-parameter quasi- $f^{\beta,\sigma}$-power increasing sequence instead of a quasi-$\beta$-power increasing sequence. The first result (Theorem 2.1) in this paper covers the case when $0<\beta <1$ and $\sigma\geqq 0$. The second main result (Theorem 2.3) in this paper covers the exceptional case when $\beta=1$ and $\sigma \leqq 0$. Each of these theorems also includes several new or known results as their special cases and consequences.