A recent note on quasi-power increasing sequence for generalized absolute summability

Volume 2009, Article ID 675403,10pages doi:10.1155/2009/675403

Research Article

A Recent Note on Quasi-Power Increasing

Sequence for Generalized Absolute Summability

E. Savas¸

and H. S¸evli

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

2Department 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 foranλnto 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_nλ_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{λn} is said to be of bounded variation bv if

n|Δλn| < ∞. 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

andenote a series with partial sums{sn}, and A a lower triangular matrix. Then anis said to be absolutelyA-summable of order k ≥ 1, written that

anis summable|A|_k, k ≥ 1, if

∞ n1

n^k−1|Tn−1− Tn|^k< ∞, 1.1

where

Tnⁿ

v0

anvsv. 1.2

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}|Tn−1− Tn|^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^−1nn.

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

Kn^βγ_n≥ m^βγ_m 1.4

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, sayγn  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^βγn} is quasi-increasing.

A positive sequenceγ  {γn} is said to be a quasi-f-power increasing sequence, if there exists a constantK  Kγ, f ≥ 1 such that Kf_nγ_n≥ f_mγ_mholds for alln ≥ m ≥ 1, 6.

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

anv ⁿ

rvanr, n, v  0, 1, . . . ,

a_nv a_nv− a_n−1,v, n  1, 2, . . . ,

1.5

where

a00 a00 a00. 1.6

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

For any matrix entryanv, Δvanv : anv− an,v 1.

Quite recently, Savas¸ 1 obtained sufficient conditions for 

anλn 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

an−1,v≥ anv forn ≥ v 1, 1.7

a_n0 1, n  0, 1, . . . , 1.8

nann O1, n −→ ∞, 1.9

n−1 v1

a_vva_{n,v 1} Oann, 1.10

m 1

nv 1

n^δk|Δ_va_nv|  O

v^δka_vv

, 1.11

m 1

nv 1

n^δkan,v 1 O v^δk

, 1.12

and let{βn} and {λn} be sequences such that

|Δλn| ≤ βn, 1.13

βn−→ 0, n −→ ∞. 1.14

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

|λn|Xn O1, n −→ ∞, 1.15

∞ n1

nX_nΔβn < ∞, 1.16

m n1

n^δk−1|sn|^k OXm, m −→ ∞, 1.17

then the series

anλnis 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 {Xn} 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 |λm|  O1, m → ∞. For example, since Xm m^−β is a quasi-β-power increasing sequence for 0 < β < 1 and if we take λm  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{λn} ∈ 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,{λn} ∈ 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 {βn} and {λn} be sequences satisfying conditions1.13 and 1.14 ofTheorem 1.1and

m n1

λn om, m −→ ∞. 2.1

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

∞ n1

nX_nβ, μ Δβn < ∞ 2.2

are satisfied, then the series 

a_nλ_n is summable |A, δ|_k, k ≥ 1, 0 ≤ δ < 1/k, where {f_n} :

{n^βlog n^μ}, μ ≥ 0, 0 ≤ β < 1, and Xnβ, μ : n^βlog n^μXn.

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 {βn} and {λn} 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

nX_nβ Δβn < ∞ 2.3

are satisfied, whereX_nβ : n^βX_n, then the seriesa_nλ_nis 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{Xn}, {βn}, 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

Φ_n:ⁿ

k1

ϕ_k, Ψ_n:^∞

kn

Δϕk. 3.1

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

ϕn ≤ 2Ψn 3.2

for alln ≥ N.

Lemma 3.2 see 8. If {Xn} is a quasi-f-increasing sequence, where {fn}  {n^βlog n^μ}, μ ≥ 0, 0 ≤ β < 1, then conditions 2.1 ofTheorem 2.1,

m n1

|Δλ_n|  om, m −→ ∞, 3.3

∞ n1

nXn β, μ

|Δ|Δλn|| < ∞, 3.4

whereX_nβ, μ  n^βlog n^μX_n, imply conditions 1.15 and

λn−→ 0, n −→ ∞. 3.5

Lemma 3.3. If {Xn} is a quasi-f-increasing sequence, where {fn}  {n^βlog 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 {Xn} be a quasi-f-increasing sequence, where {fn}  {n^βlog n^μ}, μ ≥ 0, 0 ≤ β <

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

nβnXn O1, 3.6

∞ n1

βnXn< ∞. 3.7

Proof. It is clear that if{X_n} is quasi-f-increasing, then {n^βlog n^μX_n} is quasi-increasing.

Sinceβn → 0, n → ∞, from the fact that {n^1−βlog n^−μ} is increasing and 2.2, we have

nβ_nX_n nX_n^∞

kn

Δβk

 O1n^1−β

logn _−μ^∞

kn

k^β logk _μ

X_kΔβk

 O1^∞

kn

kX_kΔβk  O1.

3.8

Again using2.2,

∞ n1

β_nX_n O1^∞

n1

X_n^∞

kn

Δβk

 O1^∞

k1

Δβk^k

n1

n^β logn _μ

Xnn^−β

logn _−μ

 O1^∞

k1

k^β logk _μ

XkΔβk^k

n1

n^−β

logn _−μ

 O1^∞

k1

kX_kβ, μ Δβk  O1.

3.9

4. Proof of Theorem 2.1

Lety_ndenote thenth term of the A-transform of the series

a_nλ_n. Then, by definition, we have

y_nⁿ

i0

a_nis_i ⁿ

v0

a_nvλ_va_v. 4.1

Then, forn ≥ 1, we have

Y_n: y_n− y_n−1ⁿ

v0

a_nvλ_va_v. 4.2

Applying Abel’s transformation, we may write

Ynⁿ⁻¹

v1

Δvanvλv^v

r1

ar annλn

n v1

av. 4.3

Since

Δ_vanvλ_v  λvΔ_va_nv Δλ_va_{n,v 1}, 4.4

we have

Yn annλnsn ⁿ⁻¹

v1

Δvanvλvsv ⁿ⁻¹

v1

an,v 1Δλvsv

 Yn,1 Yn,2 Yn,3, say.

4.5

Since

|Yn,1 Yn,2 Yn,3|^k≤ 3^k

|Yn,1|^k |Yn,2|^k |Yn,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}|Yn,1|^k^m

n1

n^{δk k−1}|annλnsn|^k

≤^m

n1

n^δknann^k−1ann|λn|^k−1|λn||sn|^k

 O1^m

n1

n^δka_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

|Δλn|ⁿ

v1

v^δkavv|sv|^k O1|λm|^m

v1

v^δkavv|sv|^k

 O1^m−1

n1

|Δλn|ⁿ

v1

v^δk−1|sv|^k O1|λm|^m

v1

v^δk−1|sv|^k

 O1^m−1

n1

β_nX_n O1|λm|X_m O1 as m −→ ∞.

4.9

Applying H ¨older’s inequality,

I2^{m 1}

n2

n^{δk k−1}|Yn,2|^k O1^{m 1}

n2

n^{δk k−1} _n−1



v1

|Δvanv||λv||sv| _k

 O1^{m 1}

n2

n^{δk k−1}ⁿ⁻¹

v1

|Δvanv||λv|^k|sv|^k _n−1

v1

|Δvanv| _k−1

.

4.10

Using1.9 and 1.11 and boundedness of {λn},

I2 O1^{m 1}

n2

n^δknann^k−1n−1

v1

|Δvanv||sv|^k|λv|^k−1|λv|

 O1^m

v1

|λ_v||s_v|^{k m 1}

nv 1

n^δk|Δ_va_nv|

 O1^m

v1

v^δka_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,

I3^{m 1}

n2

n^{δk k−1}|Yn,3|^k O1^{m 1}

n2

n^{δk k−1} _n−1

v1

an,v 1|Δλv||sv| _k

 O1^{m 1}

n2

n^{δk k−1}ⁿ⁻¹

v1

an,v 1|Δλv|^k|sv|^ka^1−k_{vv n−1}

v1

avvan,v 1

_k−1

 O1^{m 1}

n2

n^δkⁿ⁻¹

v1

a_{n,v 1}|Δλv|^k|sv|^ka^1−k_vv

 O1^m

v1

|Δλ_v|^k|s_v|^ka^1−k_vv ^{m 1}

nv 1

n^δka_{n,v 1}

 O1^m

v1

v|Δλ_v|^kv^δka_vv|s_v|^k.

4.12

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

n|Δλn| ≤ 2n^∞

kn

|Δ|Δλk|| ≤ 2^∞

knk|Δ|Δλk|| 4.13

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

1.13,

I3  O1^m

v1

v|Δλv|^k−1v|Δλv|v^δkavv|sv|^k

 O1^m

v1

vβvv^δk−1|sv|^k.

4.14

Using Abel transformation and1.17,

I3 O1^m−1

v1

Δvβv  _v



r1

r^δk−1|sr|^k 

O1mβm

m v1

v^δk−1|sv|^k

 O1^m−1

v1Δvβv Xv O1mβmX_m.

4.15

Since

Δ vβ_v

 vβ_v− v 1βv 1 vΔβ_v− β_{v 1}, 4.16

we have

I3 O1^m−1

v1

vXvΔβv O1^m−1

v1

Xv 1βv 1 O1mXmβm

 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^βlog n^μ}, μ ≥ 0, 0 ≤ β < 1, and conditions 2.2 and

m n1

1

n|s_n|^k OXm, m −→ ∞, 5.1

are satisfied, then the seriesa_nλ_nis summable|A|_k, k ≥ 1.

Proof. If we takeδ  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 series

anλnis summable|A|_k, k ≥ 1.

A weighted mean matrix, denoted byN, pn, is a lower triangular matrix with entries anv  pv/Pn, where {pn} is nonnegative sequence with p0 > 0 and Pn : _n

v0pv → ∞, as n → ∞.

Corollary 5.3. Let {pn} be a positive sequence satisfying

npn OPn, as n −→ ∞, 5.2

m 1

nv 1

n^δk p_n PnPn−1  O

v^δk Pv

, 5.3

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^βlog n^μ}, μ ≥ 0, 0 ≤ β < 1, and conditions 1.17 and 2.2

are satisfied, then the series,a_nλ_nis summable|N, p_n, δ|_kfork ≥ 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_nλ_nis summable|N, p_n, δ|_k,k ≥ 1, 0 ≤ δ < 1/k.

References

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