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A new note on absolute matrix summability

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.

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 nm1 [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,i1).
(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 , k1 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 As=( 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 , k1 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 0as n0,
(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 , k1.

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 , k1.

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 nv+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 , k1.

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 , k1.

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.

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Correspondence to Hikmet S Özarslan.

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Özarslan, H.S., Yavuz, E. A new note on absolute matrix summability. J Inequal Appl 2013, 474 (2013). https://doi.org/10.1186/1029-242X-2013-474

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Keywords

  • absolute matrix summability
  • quasi power increasing sequences
  • infinite series