On the relation between Lebesgue summability and some other summation methods

It is shown that if $$\sum_{n=1}^{N}n\left|c_{n}\right|=O(N),$$ then Lebesgue summability, $(\mathrm{C},\beta)$ summability ($\beta>0$), Abel summability, Riemann summability, and $(\gamma,\kappa)$ summability ($\kappa\geq 1$) of the series $\sum_{n=0}^{\infty}c_{n}$ are all equivalent to one another.