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Savas¸,ekremsavas@yahoo.com Received 15 May 2009; Accepted 30 July 2009 Recommended by Ramm Mohapatra We prove two theorems on|A, δ| k, k ≥ 1, 0 ≤ δ < 1/k, summability factors for an inf

Volume 2009, Article ID 675403, 10 pages

doi:10.1155/2009/675403

Research Article

A Recent Note on Quasi-Power Increasing

Sequence for Generalized Absolute Summability

E Savas¸1 and H S¸evli2

1 Department of Mathematics, ˙Istanbul Ticaret University, ¨ Usk ¨udar, 34672-˙Istanbul, Turkey

2 Department of Mathematics, Faculty of Arts & Sciences, Y ¨uz ¨unc ¨u Yıl University, 65080-Van, Turkey

Correspondence should be addressed to E Savas¸,ekremsavas@yahoo.com

Received 15 May 2009; Accepted 30 July 2009

Recommended by Ramm Mohapatra

We prove two theorems on|A, δ| k, k ≥ 1, 0 ≤ δ < 1/k, summability factors for an infinite

series by using quasi-power increasing sequences We obtain sufficient conditions fora n λ nto

be summable|A, δ| k,k ≥ 1, 0 ≤ δ < 1/k, by using quasi-f -increasing sequences.

Copyrightq 2009 E Savas¸ and H S¸evli This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 Introduction

Quite recently, Savas¸1 obtained sufficient conditions for a n λ n to be summable |A, δ| k,

k ≥ 1, 0 ≤ δ < 1/k The purpose of this paper is to obtain the corresponding result for quasi-f-increasing sequence Our result includes and moderates the conditions of his theorem with

the special caseμ  0.

A sequence{λ n } is said to be of bounded variation bv ifn |Δλ n | < ∞ Let bv0 

bv ∩ c0, where c0denotes the set of all null sequences

The concept of absolute summability of orderk ≥ 1 was defined by Flett 2 as follows Let

a ndenote a series with partial sums{s n }, and A a lower triangular matrix Thena nis said to be absolutelyA-summable of order k ≥ 1, written thata nis summable|A| k , k ≥ 1, if

∞



n1

n k−1 |T n−1 − T n|k < ∞, 1.1 where

T nn

v0

In 3, Flett considered further extension of absolute summability in which he introduced a further parameterδ The seriesa nis said to be summable|A, δ| k,k ≥ 1, δ ≥ 0,

if

∞



n1

n δk k−1 |T n−1 − T n|k < ∞. 1.3

A positive sequence{b n} is said to be an almost increasing sequence if there exist an increasing sequence{c n } and positive constants A and B such that Ac n ≤ b n ≤ Bc nsee 4 Obviously, every increasing sequence is almost increasing However, the converse need not

be true as can be seen by taking the example, sayb n  e−1n

n.

A positive sequenceγ : {γ n } is said to be a quasi-β-power increasing sequence if there

exists a constantK  Kβ, γ ≥ 1 such that

holds for alln ≥ m ≥ 1 It should be noted that every almost increasing sequence is a

quasi-β-power increasing sequence for any nonnegativeβ, but the converse need not be true as can be

seen by taking an example, sayγ n  n −βforβ > 0 see 5 If 1.4 stays with β  0, then γ is

simply called a quasi-increasing sequence It is clear that if{γ n } is quasi-β-power increasing,

then{n β γ n} is quasi-increasing

A positive sequenceγ  {γ n } is said to be a quasi-f-power increasing sequence, if there

exists a constantK  Kγ, f ≥ 1 such that Kf n γ n ≥ f m γ mholds for alln ≥ m ≥ 1, 6

We may associateA two lower triangular matrices A and  A as follows:

a nv n

rv a nr , n, v  0, 1, ,

a nv  a nv − a n−1,v , n  1, 2, ,

1.5

where

Given any sequence{x n }, the notation x n  O1 means x n  O1 and 1/x n  O1.

For any matrix entrya nv , Δ v a nv : anv − a n,v 1

Quite recently, Savas¸ 1 obtained sufficient conditions for a n λ n to be summable

|A, δ| k,k ≥ 1, 0 ≤ δ < 1/k as follows.

Theorem 1.1 Let A be a lower triangular matrix with nonnegative entries satisfying

a n−1,v ≥ a nv for n ≥ v 1, 1.7

n−1



v1

a vv a n,v 1  Oa nn , 1.10

m 1



nv 1

n δk|Δv a nv |  Ov δk a vv, 1.11

m 1

nv 1

n δk a n,v 1  Ov δk

and let {β n } and {λ n } be sequences such that

If {X n } is a quasi-β-power increasing sequence for some 0 < β < 1 such that

∞



n1

m



n1

n δk−1 |s n|k  OX m , m −→ ∞, 1.17

then the series

a n λ n is summable |A, δ| k , k ≥ 1, 0 ≤ δ < 1/k.

Theorem 1.1 enhanced a theorem of Savas 7 by replacing an almost increasing sequence with a quasi-β-power increasing sequence for some 0 < β < 1 It should be

noted that if {X n} is an almost increasing sequence, then 1.15 implies that the sequence

{λ n } is bounded However, when {X n } is a quasi-β-power increasing sequence or a

quasi-f-increasing sequence,1.15 does not imply |λ m |  O1, m → ∞ For example, since X m  m −β

is a quasi-β-power increasing sequence for 0 < β < 1 and if we take λ m  m δ , 0 < δ < β < 1,

then |λ m |X m  m δ−β  O1, m → ∞ holds but |λ m |  m δ / O1 see 8 Therefore, we remark that condition{λ n } ∈ bv0should be added to the statement ofTheorem 1.1

The goal of this paper is to prove the following theorem by using quasi-f-increasing

sequences Our main result includes the moderated version of Theorem 1.1 We will show that the crucial condition of our proof,{λ n } ∈ bv0, can be deduced from another condition

of the theorem Also, we shall eliminate condition1.15 in our theorem; however we shall deduce this condition from the conditions of our theorem

2 The Main Results

We now shall prove the following theorems

Theorem 2.1 Let A satisfy conditions 1.7–1.12, and let {β n } and {λ n } be sequences satisfying

conditions1.13 and 1.14 of Theorem 1.1 and

m



n1

If {X n } is a quasi-f-increasing sequence and conditions 1.17 and

∞



n1

are satisfied, then the series 

a n λ n is summable |A, δ| k , k ≥ 1, 0 ≤ δ < 1/k, where {f n} :

{n β log n μ }, μ ≥ 0, 0 ≤ β < 1, and X n β, μ : n β log n μ X n .

Theorem 2.1includes the following theorem with the special caseμ  0.Theorem 2.2

moderates the hypotheses ofTheorem 1.1

Theorem 2.2 Let A satisfy conditions 1.7–1.12, and let {β n } and {λ n } be sequences satisfying

conditions1.13, 1.14, and 2.1 If {X n } is a quasi-β-power increasing sequence for some 0 ≤ β < 1

and conditions1.17 and

∞



n1

are satisfied, where X n β : n β X n , then the seriesa n λ n is summable |A, δ| k , k ≥ 1, 0 ≤ δ < 1/k Remark 2.3 The crucial condition, {λ n } ∈ bv0, and condition 1.15 do not appear among the conditions of Theorems2.1and2.2 ByLemma 3.3, under the conditions on{X n }, {β n }, and {λ n} as taken in the statement ofTheorem 2.1, also in the statement ofTheorem 2.2with the special caseμ  0, conditions {λ n } ∈ bv0and1.15 hold

3 Lemmas

We shall need the following lemmas for the proof of our mainTheorem 2.1

Lemma 3.1 see 9 Let {ϕ n } be a sequence of real numbers and denote

Φn:n

k1

ϕ k , Ψn:∞

kn

IfΦn  on, then there exists a natural number N such that

for all n ≥ N.

Lemma 3.2 see 8 If {X n } is a quasi-f-increasing sequence, where {f n }  {n β log n μ }, μ ≥

0, 0 ≤ β < 1, then conditions 2.1 of Theorem 2.1 ,

m



n1

∞



n1

nX n

where X n β, μ  n β log n μ X n , imply conditions 1.15 and

Lemma 3.3 If {X n } is a quasi-f-increasing sequence, where {f n }  {n β log n μ }, μ ≥ 0, 0 ≤ β < 1,

then, under conditions1.13, 1.14, 2.1, and 2.2, conditions 1.15 and 3.5 are satisfied.

Proof It is clear that 1.13 and 1.14⇒3.3 Also, 1.13 and 2.2⇒3.4 ByLemma 3.2, under conditions1.13-1.14 and 2.1–2.2, we have 1.15 and 3.5

Lemma 3.4 Let {X n } be a quasi-f-increasing sequence, where {f n }  {n β log n μ }, μ ≥ 0, 0 ≤ β <

1 If conditions 1.13, 1.14, and 2.2 are satisfied, then

∞



n1

Proof It is clear that if {X n } is quasi-f-increasing, then {n β log n μ X n} is quasi-increasing Sinceβ n → 0, n → ∞, from the fact that {n1−βlog n −μ} is increasing and 2.2, we have

nβ n X n  nX n∞

kn

Δβ k

 O1n1−β

logn −μ∞

kn

k β logk μ X k Δβ k

 O1∞

kn

kX k Δβ k   O1.

3.8

Again using2.2,

∞



n1

β n X n  O1∞

n1

X n∞

kn

Δβ k

 O1∞

k1

Δβ kk n1

n β logn μ X n n −β

logn −μ

 O1∞

k1

k β logk μ X k Δβ kk

n1

n −β logn −μ

 O1∞

k1

kX kβ, μ Δβ k   O1.

3.9

Lety ndenote thenth term of the A-transform of the seriesa n λ n Then, by definition, we

have

y nn

i0

a ni s i n

v0

Then, forn ≥ 1, we have

Y n: yn − y n−1n

v0

Applying Abel’s transformation, we may write

Y nn−1

v1

Δv a nv λ vv

r1

a r a nn λ n

n



v1

Since

Δv a nv λ v   λ vΔv a nv Δλ v a n,v 1 , 4.4

we have

Y n  a nn λ n s n n−1

v1

Δv a nv λ v s v n−1

v1

a n,v 1 Δλ v s v

 Y n,1 Y n,2 Y n,3 , say.

4.5

|Y n,1 Y n,2 Y n,3|k≤ 3k

|Y n,1|k |Y n,2|k |Y n,3|k, 4.6

to complete the proof, it is sufficient to show that

∞



n1

n δk k−1 |Y n,r|k < ∞, for r  1, 2, 3. 4.7

Since{λ n} is bounded byLemma 3.3, using1.9, we have

I1 m

n1

n δk k−1 |Y n,1|km

n1

n δk k−1 |a nn λ n s n|k

≤m

n1

n δk na nnk−1 a nn |λ n|k−1 |λ n ||s n|k

 O1m

n1

n δk a nn |λ n ||s n|k

4.8

Using properties1.15, in view ofLemma 3.3, and3.7, from 1.9, 1.13, and 1.17,

I1 O1 m−1

n1

|Δλ n|n

v1

v δk a vv |s v|k O1|λ m|m

v1

v δk a vv |s v|k

 O1 m−1

n1

|Δλ n|n

v1

v δk−1 |s v|k O1|λ m|m

v1

v δk−1 |s v|k

 O1 m−1

n1

β n X n O1|λ m |X m  O1 as m −→ ∞.

4.9

Applying H ¨older’s inequality,

I2m 1

n2

n δk k−1 |Y n,2|k  O1 m 1

n2

n δk k−1 n−1

v1

|Δv a nv ||λ v ||s v|

k

 O1 m 1

n2

n δk k−1n−1

v1

|Δv a nv ||λ v|k |s v|k n−1

v1

|Δv a nv|

k−1

.

4.10

Using1.9 and 1.11 and boundedness of {λ n },

I2 O1 m 1

n2

n δk na nnk−1 n−1

v1

|Δv a nv ||s v|k |λ v|k−1 |λ v|

 O1m

v1

|λ v ||s v|k m 1

nv 1

n δk|Δv a nv|

 O1m

v1

v δk a vv |λ v ||s v|k  O1, as m −→ ∞,

4.11

as in the proof ofI1.

Finally, again using H ¨older’s inequality, from1.9, 1.10, and 1.12,

I3m 1

n2

n δk k−1 |Y n,3|k  O1 m 1

n2

n δk k−1 n−1

v1

a n,v 1 |Δλ v ||s v|

k

 O1 m 1

n2

n δk k−1n−1

v1

a n,v 1 |Δλ v|k |s v|k a1−k

vv n−1



v1

a vv a n,v 1

k−1

 O1 m 1

n2

n δkn−1

v1

a n,v 1 |Δλ v|k |s v|k a1−k

 O1m

v1

|Δλ v|k |s v|k a1−k

vv

m 1

nv 1

n δk a n,v 1

 O1m

v1

v|Δλ v|k v δk a vv |s v|k

4.12

ByLemma 3.1, condition3.3, in view ofLemma 3.3, implies that

n|Δλ n | ≤ 2n∞

kn

|Δ|Δλ k|| ≤ 2∞

holds Thus, byLemma 3.3,3.4 implies that {n|Δλ n|} is bounded Therefore, from 1.9 and

1.13,

I3  O1m

v1

v|Δλ v|k−1 v|Δλ v |v δk a vv |s v|k

 O1m

v1

vβ v v δk−1 |s v|k

4.14

Using Abel transformation and1.17,

I3 O1 m−1

v1

Δvβ v  v

r1

r δk−1 |s r|k

O1mβ m

m



v1

v δk−1 |s v|k

 O1 m−1

v1 Δvβ v X v O1mβ m X m

4.15

Since

Δvβ v  vβ v − v 1β v 1  vΔβ v − β v 1 , 4.16

we have

I3 O1 m−1

v1

vX v Δβ v  O1 m−1

v1

X v 1 β v 1 O1mX m β m

 O1, as m −→ ∞,

4.17

by virtue of2.2 and properties 3.6 and 3.7 ofLemma 3.4

So we obtain4.7 This completes the proof

5 Corollaries and Applications to Weighted Means

Settingδ  0 in Theorems2.1and2.2yields the following two corollaries, respectively

Corollary 5.1 Let A satisfy conditions 1.7–1.10, and let {β n } and {λ n } be sequences satisfying

conditions 1.13, 1.14, and 2.1 If {X n } is a quasi-f-increasing sequence, where {f n} :

{n β log n μ }, μ ≥ 0, 0 ≤ β < 1, and conditions 2.2 and

m



n1

1

are satisfied, then the seriesa

n λ n is summable |A| k , k ≥ 1.

Proof If we take δ  0 inTheorem 2.1, then condition1.17 reduces condition 5.1 In this case conditions1.11 and 1.12 are obtained by conditions 1.7–1.10

Corollary 5.2 Let A satisfy conditions 1.7–1.10, and let {β n } and {λ n } be sequences satisfying

conditions1.13, 1.14, and 2.1 If {X n } is a quasi-β-power increasing sequence for some 0 ≤ β < 1

and conditions2.3 and 5.1 are satisfied, then the seriesa n λ n is summable |A| k , k ≥ 1.

A weighted mean matrix, denoted byN, p n , is a lower triangular matrix with entries

a nv  p v /P n , where {p n } is nonnegative sequence with p0 > 0 and P n : n v0 p v → ∞, as

n → ∞.

Corollary 5.3 Let {p n } be a positive sequence satisfying

m 1

nv 1

n δk p n

P n P n−1  O v δk

P v

and let {β n } and {λ n } be sequences satisfying conditions 1.13, 1.14, and 2.1 If {X n } is a

quasi-f-increasing sequence, where {f n } : {n β log n μ }, μ ≥ 0, 0 ≤ β < 1, and conditions 1.17 and 2.2

are satisfied, then the series,a

n λ n is summable |N, p n , δ| k for k ≥ 1 and 0 ≤ δ < 1/k.

Proof InTheorem 2.1setA  N, p n It is clear that conditions 1.7, 1.8, and 1.10 are automatically satisfied Condition1.9 becomes condition 5.2, and conditions 1.11 and

1.12 become condition 5.3 for weighted mean method

Corollary 5.3includes the following result with the special caseμ  0.

Corollary 5.4 Let {p n } be a positive sequence satisfying 5.2 and 5.3, and let {X n } be a

quasi-β-power increasing sequence for some 0 ≤ β < 1 Then under conditions 1.13, 1.14, 1.17, 2.1,

and2.3,a n λ n is summable |N, p n , δ| k , k ≥ 1, 0 ≤ δ < 1/k.

References

1 E Savas¸, “Quasi-power increasing sequence for generalized absolute summability,” Nonlinear Analysis:

Theory, Methods & Applications, vol 68, no 1, pp 170–176, 2008.

2 T M Flett, “On an extension of absolute summability and some theorems of Littlewood and Paley,”

Proceedings of the London Mathematical Society, vol 7, pp 113–141, 1957.

3 T M Flett, “Some more theorems concerning the absolute summability of Fourier series and power

series,” Proceedings of the London Mathematical Society, vol 8, pp 357–387, 1958.

4 S Alijancic and D Arendelovic, “O-regularly varying functions,” Publications de l’Institut Math´ematique,

vol 22, no 36, pp 5–22, 1977

5 L Leindler, “A new application of quasi power increasing sequences,” Publicationes Mathematicae

Debrecen, vol 58, no 4, pp 791–796, 2001.

6 W T Sulaiman, “Extension on absolute summability factors of infinite series,” Journal of Mathematical

Analysis and Applications, vol 322, no 2, pp 1224–1230, 2006.

7 E Savas¸, “On almost increasing sequences for generalized absolute summability,” Mathematical

Inequalities & Applications, vol 9, no 4, pp 717–723, 2006.

8 H S¸evli and L Leindler, “On the absolute summability factors of infinite series involving

quasi-power-increasing sequences,” Computers & Mathematics with Applications, vol 57, no 5, pp 702–709, 2009.

9 L Leindler, “A note on the absolute Riesz summability factors,” Journal of Inequalities in Pure and Applied

Mathematics, vol 6, no 4, article 96, 5 pages, 2005.