Open Access

A new note on absolute matrix summability

Journal of Inequalities and Applications20132013:474

https://doi.org/10.1186/1029-242X-2013-474

Received: 21 November 2012

Accepted: 23 September 2013

Published: 7 November 2013

Abstract

In the present paper, we have proved theorems dealing with matrix summability factors by using quasi β-power increasing sequences. Some new results have also been obtained.

MSC:40D15, 40F05, 40G99.

Keywords

absolute matrix summabilityquasi power increasing sequencesinfinite series

1 Introduction

A positive sequence ( γ n ) is said to be quasi β-power increasing sequence if there exists a constant K = K ( β , γ ) 1 such that K n β γ n m β γ m holds for all n m 1 [1]. A sequence ( λ n ) is said to be of bounded variation, denote by ( λ n ) BV , if n = 1 | Δ λ n | = n = 1 | λ n λ n + 1 | < . Let a n be a given infinite series with the partial sums ( s n ) . Let ( p n ) be a sequence of positive numbers such that
P n = v = 0 n p v as  n ( P i = p i = 0 , i 1 ) .
(1)
The sequence-to-sequence transformation
σ n = 1 P n v = 0 n p v s v
(2)

defines the sequence ( σ n ) of the ( N ¯ , p n ) mean of the sequence ( s n ) , generated by the sequence of coefficients ( p n ) [2].

The series a n is said to be summable | N ¯ , p n | k , k 1 if [3]
n = 1 ( P n p n ) k 1 | σ n σ n 1 | k < .
(3)
Let A = ( a n v ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = ( s n ) to A s = ( A n ( s ) ) , where
A n ( s ) = v = 0 n a n v s v , n = 0 , 1 , .
(4)
The series a n is said to be summable | A , p n | k , k 1 if [4]
n = 1 ( P n p n ) k 1 | Δ ¯ A n ( s ) | k < ,
(5)
where
Δ ¯ A n ( s ) = A n ( s ) A n 1 ( s ) .

Before stating the main theorem, we must first introduce some further notations.

Given a normal matrix A = ( a n v ) , we associate two lower semimatrices A ¯ = ( a ¯ n v ) and A ˆ = ( a ˆ n v ) as follows:
a ¯ n v = i = v n a n i , n , v = 0 , 1 ,
(6)
and
a ˆ 00 = a ¯ 00 = a 00 , a ˆ n v = a ¯ n v a ¯ n 1 , v , n = 1 , 2 , .
(7)
It may be noted that A ¯ and A ˆ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then, we have
A n ( s ) = v = 0 n a n v s v = v = 0 n a ¯ n v a v
(8)
and
Δ ¯ A n ( s ) = v = 0 n a ˆ n v a v .
(9)

2 Known result

Recently, many authors have come up with theorems dealing with the applications of power increasing sequences [1, 57]. Among them, Bor and Özarslan have proved two theorems for | N ¯ , p n | k summability method by using quasi β-power increasing sequence [5]. Their theorems are as follows.

Theorem A Let ( X n ) be a quasi β-power increasing sequence for some 0 < β < 1 , and let there be sequences ( β n ) and ( λ n ) such that
| Δ λ n | β n ,
(10)
β n 0 as  n 0 ,
(11)
n = 1 n | Δ β n | X n < ,
(12)
| λ n | X n = O ( 1 ) as  n .
(13)
If
v = 1 n | s v | k v = O ( X n ) ,
(14)
n = 1 m p n P n | s n | k = O ( X m ) , m ,
(15)

then a n λ n is summable | N ¯ , p n | k , k 1 .

Theorem B Let ( X n ) be a quasi β-power increasing sequence for some 0 < β < 1 , and let sequences ( β n ) and ( λ n ) satisfy conditions (10)-(13) and (15). If
n = 1 P n | Δ β n | X n < ,
(16)
n = 1 m | s n | k P n = O ( X m ) ,
(17)

then a n λ n is summable | N ¯ , p n | k , k 1 .

3 The main result

The aim of this paper is to generalize Theorem A and Theorem B to | A , p n | k summability. Now, we shall prove the following two theorems.

Theorem 1 Let A = ( a n v ) be a positive normal matrix such that
a ¯ n 0 = 1 , n = 0 , 1 , ,
(18)
a n 1 , v a n v , for  n v + 1 ,
(19)
a n n = O ( p n P n ) ,
(20)
and ( X n ) is a quasi β-power increasing sequence for some 0 < β < 1 . If all the conditions of Theorem A and
( λ n ) BV
(21)

are satisfied, then the series a n λ n is summable | A , p n | k , k 1 .

In the special case of a n v = p v P n , this theorem reduces to Theorem A.

Theorem 2 Let A = ( a n v ) be a positive normal matrix as in Theorem 1, and let ( X n ) is a quasi β-power increasing sequence for some 0 < β < 1 . If all the conditions of Theorem B and (21) are satisfied, then the series a n λ n is summable | A , p n | k , k 1 .

We need following lemmas for the proof of our theorems.

Lemma 1 [1]

Let ( X n ) be a quasi β-power increasing sequence for some 0 < β < 1 . If conditions (11) and (12) satisfied, then
n X n β n = O ( 1 ) as  n ,
(22)
n = 1 X n β n < .
(23)
Lemma 2 Let ( X n ) be a quasi β-power increasing sequence for some 0 < β < 1 . If conditions (11) and (16) are satisfied, then
P n β n X n = O ( 1 ) ,
(24)
n = 1 p n β n X n < .
(25)

The proof of Lemma 2 is similar to that of Bor in [8] and hence is omitted.

4 Proof of Theorem 1

Let ( T n ) denote A-transform of the series a n λ n . Then by (8), (9) and applying Abel’s transformation, we have
Δ ¯ T n = v = 1 n a ˆ n v a v λ v = v = 1 n 1 Δ v ( a ˆ n v λ v ) k = 1 v a k + a ˆ n n λ n v = 1 n a v = v = 1 n 1 ( a ˆ n v λ v a ˆ n , v + 1 λ v + 1 ) s v + a n n λ n s n = v = 1 n 1 ( a ˆ n v λ v a ˆ n , v + 1 λ v + 1 a ˆ n , v + 1 λ v + a ˆ n , v + 1 λ v ) s v + a n n λ n s n = v = 1 n 1 Δ v ( a ˆ n v ) λ v s v + v = 1 n 1 a ˆ n , v + 1 Δ λ v s v + a n n λ n s n = T n , 1 + T n , 2 + T n , 3 say .
Since
| T n , 1 + T n , 2 + T n , 3 | k 3 k ( | T n , 1 | k + T n , 2 | k + T n , 3 | k ) ,
to complete the proof of the Theorem 1, it is sufficient to show that
n = 1 ( P n / p n ) k 1 | T n , r | k < , for  r = 1 , 2 , 3 .
(26)
First, applying Hölder’s inequality with indices k and k , where k > 1 and 1 k + 1 k = 1 , we get that
n = 2 m + 1 ( P n p n ) k 1 | T n , 1 | k n = 2 m + 1 ( P n p n ) k 1 ( v = 1 n 1 | Δ v a ˆ n v | | λ v | | s v | ) k = O ( 1 ) n = 2 m + 1 ( P n p n ) k 1 ( v = 1 n 1 | Δ v a ˆ n v | | λ v | k | s v | k ) × ( v = 1 n 1 | Δ v a ˆ n v | ) k 1 = O ( 1 ) n = 2 m + 1 ( P n p n a n n ) k 1 ( v = 1 n 1 | Δ v a ˆ n v | | λ v | k | s v | k ) = O ( 1 ) v = 1 m | λ v | k | s v | k n = v + 1 m + 1 | Δ v a ˆ n v | = O ( 1 ) v = 1 m p v P v | λ v | k 1 | λ v | | s v | k = O ( 1 ) v = 1 m p v P v | λ v | | s v | k = O ( 1 ) v = 1 m 1 Δ | λ v | i = 1 v p i P i | s i | k + O ( 1 ) | λ m | v = 1 m p v P v | s v | k = O ( 1 ) v = 1 m 1 β v X v + O ( 1 ) | λ m | X m = O ( 1 ) as  m ,

by virtue of the hypotheses of Theorem 1 and Lemma 1.

Since ( λ n ) BV by (21), applying Hölder’s inequality with the same indices as those above, we have
n = 2 m + 1 ( P n p n ) k 1 | T n ( 2 ) | k n = 2 m + 1 ( P n p n ) k 1 ( v = 1 n 1 | Δ λ v | | a ˆ n , v + 1 | | s v | ) k = O ( 1 ) n = 2 m + 1 ( P n p n ) k 1 ( v = 1 n 1 | Δ λ v | | a ˆ n , v + 1 | | s v | k ) × ( v = 1 n 1 | Δ λ v | | a ˆ n , v + 1 | ) k 1 = O ( 1 ) n = 2 m + 1 ( P n p n a n n ) k 1 ( v = 1 n 1 β v | a ˆ n , v + 1 | | s v | k ) × ( v = 1 n 1 | Δ λ v | ) k 1 = O ( 1 ) v = 1 m β v | s v | k n = v + 1 m + 1 | a ˆ n , v + 1 | = O ( 1 ) v = 1 m β v | s v | k = O ( 1 ) v = 1 m ( v β v ) | s v | k v = O ( 1 ) v = 1 m 1 Δ ( v β v ) i = 1 v | s i | k i + O ( 1 ) m β m v = 1 m | s v | k v = O ( 1 ) v = 1 m 1 Δ ( v β v ) X v + O ( 1 ) m β m X m = O ( 1 ) v = 1 m 1 v | Δ β v | X v + O ( 1 ) v = 1 m 1 β v + 1 X v + 1 + O ( 1 ) m β m X m = O ( 1 ) as  m ,

by virtue of the hypotheses of Theorem 1 and Lemma 1.

Finally, by following the similar process as in T n , 1 , we have that
n = 1 m ( P n p n ) k 1 | T n ( 3 ) | k n = 1 m ( P n p n ) k 1 | a n n | k | λ n | k | s n | k = O ( 1 ) n = 1 m p n P n | λ n | | s n | k = O ( 1 ) as  m .
So, we get
n = 1 ( P n / p n ) k 1 | T n , r | k < , for  r = 1 , 2 , 3 .

This completes the proof of Theorem 1.

5 Proof of Theorem 2

Using Lemma 2 and proceeding as in the proof of Theorem 1, replacing v = 1 m β v | s v | k by v = 1 m β v P v ( | s v | k P v ) , we can easily prove Theorem 2.

If we take p n = 1 in these theorems, then we have two new results dealing with | A | k summability factors of infinite series. Also, if we take k = 1 , then we obtain another two new results concerning | A | summability. Finally, by taking ( X n ) as almost increasing sequence in the theorems, we get new results dealing with | A , p n | k summability factors of infinite series.

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Erciyes University

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Copyright

© Özarslan and Yavuz; licensee Springer. 2013

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