On a result of Flett for Cesلro matrices

Let s n be the partial sums of the series ∑ n = 0 ∞ a n . We consider the sufficient conditions for a matrix T = ( t n k ) such that ∑ n = 1 ∞ α n | s n − s n − 1 | k < ∞ implies ∑ n = 1 ∞ β n | t n − t n − 1 | s < ∞ where { α n } and { β n } are two given positive sequences and k , s > 0 , and { t n } is the T - transformation of { s n } . Our results extend the related results of Flett [T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc. 7 (1957) 113–141] and Savas and Sevli [E. Savaş, H. Sevli, On extension of a result of Flett for Cesáro matrices, Appl. Math. Lett. 20 (2007) 476–478].