The well-known classical Tauberian theorems given for A_{\lambda} (the discrete Abel mean) by Armitage and Maddox in [Armitage, H. D and Maddox, J. I., Discrete Abel means, Analysis, 10 (1990), 177–186] is generalized. Similarly the “one-sided” Tauberian theorems of Landau and Schmidt for the Abel method are extended by replacing \lim As with Abel-\lim A\sigma_{n}^{i}(s).
Slowly oscillating of \{s_{n}\} is a Tauberian condition of the Hardy-Littlewood Tauberian theorem for Borel summability which is also given by replacing \lim_{t} (Bs)_{t} = \ell, where t is a continuous parameter,
with \lim_{n}(Bs)_{n} = \ell, and further replacing it by Abel-\lim (B\sigma_{k}^{i}(s))_{n} = \ell, where B is the Borel matrix method.



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Albayrak, Mehmet, Gül, Erdal