On Inclusion Relations for Absolute Summability

We obtain necessary and (different) sufficient conditions for a series summable | ¯ N, p n | k , 1 < k ≤ s < ∞, to imply that the series is summable |T | s , where (¯ N, p n) is a weighted mean matrix and T is a lower triangular matrix. As corollaries of this result, we obtain several inclusion theorems. Let a n be a given series with partial sums s n , (C, α) the Césaro matrix of order α. If σ α n denotes the nth term of the (C, α)-transform of {s n } then, from Flett [4], a n is said to be summable |C, α| k , k ≥ 1 if ∞ n=1 n k−1 σ α n − σ α n−1 k < ∞. (1) For any sequence {u n }, the forward difference operator ∆ is defined by ∆u n = u n − u n+1. An appropriate extension of (1) to arbitrary lower triangular matrices T is ∞ n=1 n k−1 ∆t n−1 k < ∞, (2) where t n := n k=0 t nk s k. (3) Such an extension is used, for example, in Bor [2]. But Sarigöl, Sulaiman, and Bor and Thorpe [3] make the following extension of (1). They define a series a n to be summable | ¯ N, p n | k , k ≥ 1 if ∞ n=1 P n p n k−1 ∆Z n−1 k < ∞, (4) where Z n denotes the nth term of the weighted mean transform of {s n }; that is, Z n = 1 P n n k=0 p k s k. Apparently they have interpreted the n in (1) to represent the reciprocal of the nth diagonal term of the matrix (¯ N, p n). (See, e.g., Sarigöl [6], where this is explicitly the case.)