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, $(\mathrm{C},β)$ summability ($β>0$), Abel summability, Riemann summability, and $(γ,κ)$ summability ($κ\geq 1$) of the series $\sum_{n=0}^{\infty}c_{n}$ are all equivalent to one another.

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