Some remarks on summability factors

Abstract:

Bosanquet [2] showed that a necessary and sufficient condition for $\Sigma _{k = 1}^\infty {x_k}{y_k}$ to be Cesàro summable of order $n$ ($n$ is a nonnegative integer) whenever $\sigma _k^n(y) = o(k)$ where $\sigma _k^n(y)$ is the $k$ th Cesàro mean of $y$ of order $n$ is that $\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| < \infty$ and ${\lim _{k \to 0}}k{x_k} = 0$. The main result of this paper is to show that a necessary and sufficient condition for $\Sigma _{k = 1}^\infty {x_k}{y_k}$ to be Cesàro summable of order $n$ ($n$ is a nonnegative integer) whenever $\Sigma _{k = 1}^\infty {k^{n + 1}}|{\Delta ^{n + 1}}{x_k}| < \infty$ and ${\lim _{k \to \infty }}k{x_k} = 0$ is that $\sigma _k^n(y) = o(k)$.