In this paper, we introduce a new type of slow oscillation and slow decrease conditions. We prove that these or their variants are Tauberian conditions from s mn st → s to s mn → s. We also prove that they are Tauberian conditions from t 11 mn st → s to s mn → s, where t 11 mn are the weighted means of the double sequence {s mn } ∞ m,n=0 . Our results not only generalize well-known results, but also solve the conjecture of Móricz posed in [F. Móricz, Tauberian theorems for double sequences that are statistically summable (C, 1, 1),

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In this paper, the asymptotic behavior of a trigonometric series with O-regularly varying quasimonotone coefficients is investigated. Our results generalize the work Ž Ž . done by S. Aljancic, R. Bojanic, and M.

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