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.)

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 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 where 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, where Z n denotes the nth term of the weighted mean transform of {s n }; that is, 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.)Unfortunately, this interpretation cannot be correct.For if it were, then, since the nth diagonal entry of (C, α) is O(n −α ), (1) would take the form However, Flett stays with (1).Thus ( 2) is an appropriate extension of (1) to lower triangular matrices.
Given any lower triangular matrix T , we can associate the matrices T and T with entries defined by respectively.Thus, from (3), For a weighted mean matrix A = ( N, p n ), so that, from (5), since P −1 = 0. We will always assume that {p n } is a positive sequence with P n → ∞.Also, ∆ ν tnν := tnν − tn,ν+1 .
Let T be a lower triangular matrix.Then, the necessary conditions for a n summable Proof.We are given that Now, the space of sequences {a n } satisfying ( 14) is a Banach space if normed by We also consider the space of those sequences {Y n } that satisfy (13).This is also a BK-space with respect to the norm Observe that (8) transforms the space of sequences satisfying (14) into the space of sequences satisfying (13).Applying the Banach-Steinhaus theorem, there exists a constant K > 0 such that Applying ( 11) and (8) to a ν = e ν − e ν+1 , where e ν is the νth coordinate vector, we have, respectively, By ( 15) and ( 16), it follows that recalling that tνν = tνν = t νν .
Using ( 19) and ( 20) in ( 17), along with (12), it follows that The above inequality will be true if and only if each term on the left-hand side is Taking the first term, which verifies that (i) is necessary.
Using the second term, we have which is condition (ii).
Corollary 2. Let T be a lower triangular matrix, {p n } satisfying (12).Then the necessary conditions for To prove Corollary 2, simply set s = k in Theorem 1.A lower triangular matrix T is called a triangle if each t nn ≠ 0.
Let T be a triangle with bounded entries such that T and {p n } satisfy the following: Proof.Solving (11) for {a n } and substituting into (8) give From Minkowski's inequality, it is sufficient to show that Using condition (i) of Theorem 3, But . Using Hölder's inequality and conditions (i), (ii), (iii), and (iv) of Theorem 3.
By Hölder's inequality and conditions (v), (vi), and (iii) of Theorem 3, we have Corollary 4 (see [5]).Let T be a nonnegative lower triangular matrix, {p n } a positive sequence satisfying Proof.Since s = k and T is nonnegative, condition (ii) of Theorem 3 is automatically satisfied, and conditions (ii), (iv), (v), and (vi) of Corollary 4 are equivalent to conditions (i), (v), (iv), and (vi) of Theorem 3, respectively (33) Therefore, using conditions (i) and (iii) of Corollary 4, and condition (iii) of Theorem 3 is satisfied.
Remark 5.For 1 < k ≤ s < ∞, necessary and sufficient conditions for a triangle A : k → s are known only for factorable matrices (see Bennett [1]), which include weighted mean matrices.Therefore, we should not expect to obtain a set of necessary and sufficient conditions when an arbitrary triangle is involved.
However, necessary and sufficient conditions for a matrix A : → s , 1 ≤ s < ∞ are known.The following result comes from Theorem 2.1 of Rhoades and Savaş [5] Remark 7. In [5], it is assumed that T has nonnegative entries and row sums one, but these restrictions are not used in the proofs.Proof.Note that, with k = 1, (12) is automatically satisfied.Therefore, the necessity of the conditions follows from Theorem 1.
To prove the conditions sufficient, use [

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n summable |C, 1| implies a n | N, q n | if and only if (i) nq n /Q n = O(1).