A note on generalized vertical bar a vertical bar(K)- summability factors for infinite series

Volume 2010, Article ID 814974,10pages doi:10.1155/2010/814974

Research Article

A Note on Generalized |A|

-Summability Factors for Infinite Series

Ekrem Savas¸

Department of Mathematics, Istanbul Ticaret University, Uskudar, 34672 Istanbul, Turkey

Correspondence should be addressed to Ekrem Savas¸,[email protected] Received 9 September 2009; Revised 9 November 2009; Accepted 14 December 2009 Academic Editor: Martin Bohner

Copyrightq 2010 Ekrem Savas¸. 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.

A general theorem concerning the|A|_k—summability factors of infinite series has been proved.

1. Introduction

A weighted mean matrix, denoted byN, p_n, is a lower triangular matrix with entries p_k/P_n, where{p_k} is a nonnegative sequence with p0> 0, and P_n:_n

k0p_k.

Mishra and Srivastava 1 obtained sufficient conditions on a sequence {pk} and a sequence{λ_n} for the seriesa_nP_nλ_n/np_nto be absolutely summable by the weighted mean matrixN, pn.

Recently Savas¸ and Rhoades2 established the corresponding result for a nonnegative triangle, using the correct definition of absolute summability of orderk ≥ 1.

Let A be an infinite lower triangular matrix. We may associate with A two lower triangular matricesA and A, whose entries are defined by

a_nk ⁿ

ik

a_ni, a_nk a_nk− a_n−1,k, 1.1

respectively. The motivation for these definitions will become clear as we proceed.

LetA be an infinite matrix. The series

akis said to be absolutely summable byA, of orderk ≥ 1, written as |A|k, if

∞ k0

n^k−1|Δt_n−1|^k< ∞, 1.2

where Δ is the forward difference operator and tn denotes the nth term of the matrix transform of the sequence{s_n}, where s_n:_n

k1a_k. Thus

tnⁿ

k1

ankskⁿ

k1

ank

k ν1

aνⁿ

ν1

aν

n kν

ank ⁿ

ν1

anνaν,

t_n− t_n−1ⁿ

ν1

a_nνa_ν−ⁿ⁻¹

ν1

a_n−1,νa_νⁿ

ν1

a_nνa_ν,

1.3

sincea_n−1,n 0.

A sequence{λ_n} is said to be of bounded variation bv if

n|Δ λ_n| < ∞. Let bv0  bv ∩ c0, where c0denotes the set of all null sequences.

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_n, see 3.

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^βγn≥ m^βγm 1.4

holds for alln ≥ m ≥ 1. It should be noted that every almost increasing sequence is quasi β-power increasing sequence for any nonnegative β, but the converse need not be true as can be seen by taking an example, sayγn  n^−βforβ > 0 see 4. If 1.4 stays with β  0 then γ is simply called a quasi-increasing sequence. It is clear that if{γn} is quasi β-power increasing then{n^βγ_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 Kfnγn≥ fmγmholds for alln ≥ m ≥ 1, where f : {f_n}  {n^βlog n^μ}, μ > 0, 0 < β < 1 was considered instead of n^βsee 5,6.

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

Quite recently, Savas¸ and Rhoades 2 proved the following theorem for |A|k- summability factors of infinite series.

Theorem 1.1. Let A be a triangle with nonnegative entries satisfying

i an0 1, n  0, 1, . . . ,

ii a_n−1,ν≥ a_nνforn ≥ ν 1,

iii nann O1,

iv Δ1/a_nn  O1, and

v_n

ν0aνν|an,ν1|  Oann.

If{X_n} is a positive nondecreasing sequence and the sequences {λ_n} and {β_n} satisfy

vi |Δλ_n| ≤ β_n,

vii lim βn 0,

viii |λ_n|X_n  O1,

ix_∞

n1nXn|Δβn| < ∞, and

x T_n:_n

ν1|s_ν|^k

ν  OX_n, then the series_∞

n1anλn/nannis summable|A|k, k ≥ 1.

It should be noted that if{X_n} is an almost increasing sequence then viii implies that the sequence{λ_n} is bounded. However, when {X_n} is a quasi β-power increasing sequence or a quasif-increasing sequence, viii does not imply |λm|  O1, m → ∞. For example, sinceX_m  m^−β is a quasiβ-power increasing sequence for 0 < β < 1, 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 7.

The goal of this paper is to prove a theorem by using quasif-increasing sequences.

We show that the crucial condition of our proof,{λ_n} ∈ bv0, can be deduced from another condition of the theorem.

2. The Main Results

We have the following theorem:

Theorem 2.1. Let A be nonnegative triangular matrix satisfying conditions i–v and let {β_n} and {λ_n} be sequences satisfying conditions (vi) and (vii) ofTheorem 1.1and

m n1

λ_n om, m −→ ∞. 2.1

If{Xn} is a quasi f-increasing sequence and condition (x) and

∞ n1

nX_nβ, μΔβn < ∞ 2.2

are satisfied, then the series_∞

n1a_nλ_n/na_nnis summable|A|_k,k ≥ 1, where {f_n} : {n^βlog n^μ}, μ ≥ 0, 0 ≤ β < 1, and X_nβ, μ : n^βlog n^μX_n.

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

Theorem 2.2. Let A satisfying conditions i–v and let {β_n} and {λ_n} be sequences satisfying conditions (vi), (vii), and2.1. If {X_n} is a quasi β-power increasing sequence for some 0 ≤ β < 1 and conditions (x) and

∞ n1

nXnβΔβn < ∞ 2.3

are satisfied, whereX_nβ : n^βX_n, then the series_∞

ν1a_nλ_n/na_nnis summable|A|_k, k ≥ 1.

If we take that {Xn} is an almost increasing sequence instead of a quasi β-power increasing sequence then ourTheorem 2.2reduces to8, Theorem 1.

Remark 2.3. The crucial condition,{λn} ∈ bv0, and condition viii do not appear among the conditions of Theorems2.1and2.2. ByLemma 3.3, under the conditions on{X_n}, {β_n}, and {λn} as taken in the statement of theTheorem 2.1, also in the statement ofTheorem 2.2with the special caseμ  0, conditions {λn} ∈ bv0andviii hold.

3. Lemmas

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

Lemma 3.1 see 9. Let {ϕn} be a sequence of real numbers and denote

Φ_n:ⁿ

k1

ϕ_k, Ψ_n:^∞

kn

Δϕk. 3.1

IfΦ_n on then there exists a natural number N such that

ϕn ≤ 2Ψn 3.2

for alln ≥ N.

Lemma 3.2 see 7. 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

nX_n β, μ

|Δ|Δλ_n|| < ∞, 3.4

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

λ_n−→ 0, n −→ ∞. 3.5

Lemma 3.3 see 7. If {Xn} is a quasi f-increasing sequence, where {fn}  {n^βlog n^μ}, μ ≥ 0, 0≤ β < 1, then under conditions (vi), (vii), 2.1 and 2.2, conditions viii and 3.5 are satisfied.

Lemma 3.4 see 7. Let {X_n} be a quasi f-increasing sequence, where {f_n}  {n^βlog n^μ}, μ ≥ 0, 0≤ β < 1. If conditions (vi), (vii), and 2.2 are satisfied, then

nβnXn O1, 3.6

∞ n1

βnXn< ∞. 3.7

4. Proof of Theorem 2.1

Let T_n denote the nth term of the A-transform of the partial sums of the series

_∞

n1a_nλ_n/na_nn. Then, we have

T_nⁿ

ν1

a_nν^ν

i1

aiλi

a_iii ^m

i1

aiλi

a_iii

n νi

a_nνⁿ

i1

a_niaiλi

a_iii. 4.1

Thus,

Tn− Tn−1ⁿ

i1

aniaiλi

aiii −ⁿ⁻¹

i1

an−1,iaiλi

aiii

ⁿ

i1

ani− a_n−1,ia_iλ_i aiii ⁿ

i1

a_nia_iλ_i aiii

ⁿ

i1

a_ni λi

a_iiis_i− s_i−1

ⁿ⁻¹

i1

a_ni λ_i

a_iiis_i a_nn λ_n

a_nnns_n−ⁿ

i1

a_niλ_is_i−1 a_iii

ⁿ⁻¹

i1

a_ni λ_i

aiiis_i a_nn λ_n

annns_n−ⁿ⁻¹

i1

a_n,i1 λ_i1s_i

i 1a

ⁿ⁻¹

i1



a_ni λ_i

aiii− a_n,i1 λ_i1

i 1a

s_i a_nn λ_n nann.

4.2

It is easy to see that

a_niλ_i

ia_ii − an,i1λi1

i 1a  Δ_i

a_ni ia_ii

λ_i an,i1

i 1a Δλi. 4.3

Also we may write

Δi

ani

ia_ii

λi Δianiλi

ia_ii an,i1λi

 1

ia_ii − 1

i 1a

. 4.4

Therefore, forn > 1,

Tn− Tn−1ⁿ⁻¹

i1

Δiani

ia_ii λisiⁿ⁻¹

i1

an,i1λi

 1

ia_ii − 1

i 1a

si

ⁿ⁻¹

i1

a_n,i1

i 1a Δiλisiλn

nsn

 Tn1 Tn2 Tn3 Tn4, say.

4.5

To complete the proof of the theorem, it will be sufficient to show that

∞ n1

n^k−1|T_nr|^k< ∞, for r  1, 2, 3, 4. 4.6

Using H ¨older’s inequality and conditioniii,

I1^m1

n1

n^k−1|Tn1|^k≤^m1

n1

n^k−1 _n−1



i1

Δ_iani iaii λisi



_k

 O1^m1

n1

n^k−1 _n−1

i1

|Δianiλisi| _k

 O1^m1

n1

n^k−1 _n−1

i1

|Δiani||λi|^k|si|^k 

× _n−1

i1

|Δiani|

_k−1 .

4.7

Sinceλn is bounded byLemma 3.3, usingii, iii, vi, x, and property 3.7 of Lemma 3.4,

I1 O1^m1

n1

nann^k−1ⁿ⁻¹

i1

|λi|^k|si|^k|Δiani|

 O1^m1

n1

na_nn^k−1 _n−1



i1

|λ_i|^k−1|λ_i||Δ_ia_ni||s_i|^k 

 O1^m

i1

|λi||si|^km1

ni1

nann^k−1|Δiani|

 O1^m

i1

|λi||si|^kaii  O1^m

i1

|λi||si|^k i

 O1

_m

i1

|λ_i|ⁱ

r1

|s_r|^k r −^m−1

i0

|λ_i1|ⁱ

r1

|s_r|^k r

 O1^m−1

i1Δ|λi|ⁱ

r1

1

r|sr|^k O1|λm|^m

i1

|si|^k i

 O1^m−1

i1

Δ|λi|Xi O1|λm|Xm

 O1^m

i1

βiXi O1|λm|Xm O1.

4.8

Now

I2^m1

n1

n^k−1|T_n2|^k^m1

n1

n^k−1



n−1 i1

a_n,i1λ_iΔ

 1 ia_ii

s_i



k

 O1^m1

n1

n^k−1

_n−1



i1

|a_n,i1||λ_i|

Δ 1 iaii

|sⁱ|

_k .

4.9

From2,

Δ

 1 ia_ii

 1

i 1

 Δ

 1 a_ii

1

ia_ii



. 4.10

Thus, usingiv and ii,

Δ 1 iaii

 

 1 i 1

 Δ

 1 aii

1

iaii



 1

i 1O1 O1.

4.11

Hence, using H ¨older’s inequality,v, iii, and the fact that the λn’s are bounded,

I2 O1^m1

n1

n^k−1

_n−1

i1

|a_n,i1||λ_i| 1 i 1|s_i|

_k

 O1^m1

n1

n^k−1

_n−1



i1

|a_n,i1|a_ii|λ_i||s_i|

_k

 O1^m1

n1

n^k−1 _n−1



i1

|an,i1|aii|λi|^k|si|^k _n−1



i1

aii|an,i1| _k−1

 O1^m1

n1

na_nn^k−1n−1

i1

|a_n,i1|a_ii|λ_i|^k|s_i|^k

 O1^m

i1

|λi|^k|si|^kaii m1

ni1

nann^k−1|an,i1|

 O1^m

i1

|λi|^k|si|^ka_ii^m1

ni1

|an,i1|

 O1^m

i1

|λi|^k|si|^kaii

 O1^m

i1

|λi||λi|^k−1|si|^k1 i

^m

i1

|λi||s_i|^k

i  O1,

4.12

as in the proof ofI1.

It follows from3.6 that βn  O1/n and hence that |Δλn|  O1/n by condition

vi.

Usingiii, H¨older’s inequality, and v,

I3^m1

n1

n^k−1|Tn3|^k^m1

n1

n^k−1



n−1

i1

a_n,i1Δλ_is_i

i 1a





k

 O1^m1

n1

n^k−1 _n−1

i1

|a_n,i1||Δλ_i||s_i| _k

 O1^m1

n1

n^k−1

_n−1



i1

a_ii

aii|a_n,i1||Δλ_i||s_i|

_k

 O1^m1

n1

n^k−1

_n−1



i1

aii|an,i1|

a^k_ii |Δλi|^k|si|^k

_n−1



i1

aii|an,i1|

_k−1

 O1^m1

n1

na_nn^k−1ⁿ⁻¹

i1

a_ii|an,i1|

a^k_ii |Δλ_i|^k|s_i|^k

 O1^m1

n1 n−1

i1

|an,i1||Δλi|^k|si|^k 1 a^k_iiaii

 O1^m

i1

a_ii

a^k_ii|Δλi|^k|si|^km1

ni1

|an,i1|

 O1^m

i0

|Δλi| a_ii

_k−1

|Δλ_i||s_i|^k

 O1^m

i1

|Δλ_i| |s_i|^k O1^m

i0

|s_i|^kβ_i.

4.13

Since|si|^k iTi− Ti−1 by x, we have

I3 O1^m

i1iTi− Ti−1βi. 4.14

Using Abel’s transformation,vi, 2.2, and properties 3.7 and 3.6 ofLemma 3.4,

I3 O1^m−1

i1

T_iΔ iβ_i

O1mTnβ_n

 O1^m−1

i1

iΔβiXi O1^m−1

i1

Xiβi O1mXnβn O1.

4.15

Using the boundedness ofλnandx,

I4 ^m1

n1

n^k−1|T_n4|^k^m1

n1

n^k−1

snλn

n

^k

^m1

n1

|sn|^k|λn|^k1 n ^m1

n1

|sn|^k

n |λn||λn|^k−1 O1,

4.16

as in the proof ofI1.

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

i0pi → ∞, as n → ∞.

Corollary 4.1. Let {pn}be a positive sequence satisfying

i np_n OP_n and

ii ΔP_n/p_n  O1.

and let{βn} and {λn} be sequences satisfying conditions vi, vii, and 2.1. If {Xn} is a quasi f-increasing sequence, where {f_n} : {n^βlog n^μ}, μ ≥ 0, 0 ≤ β < 1, and conditions (x) and 2.2

are satisfied, then the series_∞

n1anPnλn/npn is summable |N, pn|k,k ≥ 1.

Acknowledgment

The author wishes to thank the referees for their careful reading of the manuscript and for their helpful suggestions.

References

1 K. N. Mishra and R. S. L. Srivastava, “On |N, pn| summability factors of infinite series,” Indian Journal of Pure and Applied Mathematics, vol. 15, no. 6, pp. 651–656, 1984.

2 E. Savas¸ and B. E. Rhoades, “A note on |A|_ksummability factors for infinite series,” Journal of Inequalities and Applications, vol. 2007, Article ID 86095, 8 pages, 2007.

3 S. Alijancic and D. Arendelovic, “O-regularly varying functions,” Publications de l’Institut Math´ematique, vol. 22, no. 36, pp. 5–22, 1977.

4 L. Leindler, “A new application of quasi power increasing sequences,” Publicationes Mathematicae Debrecen, vol. 58, no. 4, pp. 791–796, 2001.

5 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.

6 H. S¸evli, “General absolute summability factor theorems involving quasi-power-increasing sequences,” Mathematical and Computer Modelling, vol. 50, no. 7-8, pp. 1121–1127, 2009.

7 E. Savas¸ and H. S¸evli, “A recent note on quasi-power increasing sequence for generalized absolute summability,” Journal of Inequalities and Applications, vol. 2009, Article ID 675403, 10 pages, 2009.

8 S. Kr. Saxena, “A note on summability factors for a triangular matrix,” International Journal of Mathematical Analysis, vol. 2, no. 21–24, pp. 1103–1110, 2008.

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.

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