Necessary and sufficient conditions for inclusion relations for absolute summability
نویسنده
چکیده
In this paper we obtain appropriate necessary and sufficient conditions for |N, pn|k summability to imply that of |N,qn|s for 1< k≤ s<∞. As in [6] we make use of a result of Bennett [1], who has obtained necessary and sufficient conditions for a factorable matrix to map lk → ls. A factorable matrix A is one in which each entry ank = bnck. Weighted mean matrices are factorable. It will not be possible to extended our result by replacing (N,qn) by a triangular matrix A, since necessary and sufficient conditions are not known for an arbitrary triangular matrix B to map: lk → ls. However, if k = 1, then the necessary and sufficient conditions can be obtained. Such a result is the special case, by setting each λn = 1 of Theorem 2.1 of Rhoades and Savaş [4].
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