Abstract
The goal of this paper is to obtain sufficient and (different) necessary conditions for a series , which is absolutely summable of order by a triangular matrix method , , to be such that is absolutely summable of order by a triangular matrix method . As corollaries, we obtain two inclusion theorems.
Highlights
In the recent papers [1, 2], the author obtained necessary and sufficient conditions for a series an which is absolutely summable of order k by a weighted mean method, 1 < k ≤ s < ∞, to be such that anλn is absolutely summable of order s by a triangular matrix method
Let xn denote the nth term of the A-transform of a series an, as in (6), n
Since A is a weighted mean matrix, A is a factorable triangle and it is easy to show that its inverse is bidiagonal
Summary
In the recent papers [1, 2], the author obtained necessary and sufficient conditions for a series an which is absolutely summable of order k by a weighted mean method, 1 < k ≤ s < ∞, to be such that anλn is absolutely summable of order s by a triangular matrix method. Let {λn} be a sequence of constants, A and B triangles satisfying (i) |bnnλn|/|ann| = O(ν1/s−1/k), (ii) (n|Xn|)s−k = O(1), (iii) |ann − an+1,n| = O(|annan+1,n+1|), (iv) O(|bnn λn+1|), (vii) ∞n=ν+1(n|bnnλn+1|)s−1|bn,ν+1λν+1| = O((ν|bννλν+1|)s−1), (viii)
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