On a result of Flett for Cesáro matrices

Let snsn be the partial sums of the series ∑n=0∞an. We consider the sufficient conditions for a matrix T=(tnk)T=(tnk) such that ∑n=1∞αn|sn−sn−1|k<∞ implies ∑n=1∞βn|tn−tn−1|s<∞ where {αn}{αn} and {βn}{βn} are two given positive sequences and k,s>0,and {tn}{tn} is the TT- transformation of {sn}{sn}. 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].