A summability factor theorem for quasi-power-increasing sequences

Volume 2010, Article ID 105136,10pages doi:10.1155/2010/105136

Research Article

A Summability Factor Theorem for Quasi-Power-Increasing Sequences

E. Savas¸

Department of Mathematics, ˙Istanbul Ticaret University, ¨Usk ¨udar, 34378 ˙Istanbul, Turkey

Correspondence should be addressed to E. Savas¸,ekremsavas@yahoo.com Received 23 June 2010; Revised 3 September 2010; Accepted 15 September 2010 Academic Editor: J. Szabados

Copyrightq 2010 E. 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.

We establish a summability factor theorem for summability|A, δ|_k, whereA is lower triangular matrix with nonnegative entries satisfying certain conditions. This paper is an extension of the main result of the work by Rhoades and Savas¸2006 by using quasi f-increasing sequences.

1. Introduction

Recently, Rhoades and Savas¸1 obtained sufficient conditions for

anλnto be summable

|A, δ|_k,k ≥ 1 by using almost increasing sequence. The purpose of this paper is to obtain the corresponding result for quasif-increasing sequence.

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.

LetA be a lower triangular matrix, {s_n} a sequence. Then

A_n:ⁿ

ν0

a_nνs_ν. 1.1

A series

an, with partial sumssn, is said to be summable |A|k, k ≥ 1 if

∞ n1

n^k−1|An− A_n−1|^k< ∞, 1.2

and it is said to be summable|A, δ|k, k ≥ 1 and δ ≥ 0 if see, 2

∞ n1

n^{δk k−1}|An− An−1|^k< ∞. 1.3

A positive sequence{bn} 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, 3.

Obviously, every increasing sequence is almost increasing. However, the converse need not be true as can be seen by taking the example, saybn  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. A sequence satisfying 1.4 for β  0 is 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 ≥ fmγm holds for alln ≥ m ≥ 1, where f : {f_n}  {n^βlog n^μ}, μ > 0, 0 < β < 1, see, 5.

We may associate withA 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 that x_n  O1 and 1/x_n  O1. For any matrix entry anv, Δvanv : anv− an,v 1.

Rhoades and Savas¸1 proved the following theorem for |A, δ|_ksummability factors of infinite series.

Theorem 1.1. Let {Xn} be an almost increasing sequence and let {βn} and {λn} be sequences such that

i |Δλn| ≤ βn,

ii lim βn 0,

iii_∞

n1n|Δβn|Xn< ∞,

iv |λn|Xn  O1.

LetA be a lower triangular matrix with nonnegative entries satisfying

v na_nn O1,

vi a_n−1,ν≥ a_nνforn ≥ ν 1,

vii a_n0 1 for all n,

viii_n−1

ν1aννanν 1 Oann,

ix_{m 1}

nν 1n^δk|Δνanν|  Oν^δkaνν and

x_{m 1}

nν 1n^δkanν 1 Oν^δk.

If

xi_m

n1n^δk−1|t_n|^k OX_m, where t_n: 1/n 1_n

k1ka_k, then the series

anλnis summable|A, δ|k, k ≥ 1.

It should be noted that, if{X_n} is an almost increasing sequence, then condition iv

implies that the sequence{λ_n} is bounded. However, if {X_n} is a quasi β-power increasing sequence or a quasi f-increasing sequence, iv does not imply that λ is bounded. For example, the sequence{Xm} defined by Xm  m^−β is trivially a quasiβ-power increasing sequence for eachβ > 0. If λ  {m^δ}, for any 0 < δ < β, then λ_mX_m m^δ−β O1, but λ is not bounded,see, 6,7.

The purpose 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 now will prove the following theorems.

Theorem 2.1. Let A satisfy conditions (v)–(x) and let {β_n} and {λ_n} be sequences satisfying conditions (i) and (ii) ofTheorem 1.1and

m n1

λn om, m −→ ∞. 2.1

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

∞ n1

nXnβ, μΔβn < ∞ 2.2

are satisfied then the series

anλn is summable|A, δ|_k,k ≥ 1, where {fn} : {n^βlog n^μ}, μ ≥ 0, 0 ≤ β < 1, and X_nβ, μ : n^βlog n^μX_n.

The following theorem is the special case ofTheorem 2.1forμ  0.

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

∞ n1

nX_nβΔβn < ∞ 2.3

are satisfied, whereX_nβ : n^βX_n, then the seriesa_nλ_nis summable|A, δ|_k,k ≥ 1.

Remark 2.3. The conditions {λn} ∈ bv0, and iv 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 of theTheorem 2.1, also in the statement ofTheorem 2.2with the special case μ  0, conditions {λn} ∈ bv0andiv hold.

3. Lemmas

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

Lemma 3.1 see 8. 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 9. 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 (iv) 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 (i), (ii), 2.1, and 2.2, conditions (iv) 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 (i), (ii), and 2.2 are satisfied, then

nβnXn O1, 3.6

∞ n1

βnXn< ∞. 3.7

4. Proof of Theorem 2.1

Proof. Lety_n be the nth term of the A transform of the partial sums of_n

i0λ_ia_i. Then we have

y_n:ⁿ

i0

a_nis_iⁿ

i0

a_niⁱ

ν0

λ_νa_ν

ⁿ

ν0

λ_νa_νⁿ

iν

a_niⁿ

ν0

a_nνλ_νa_ν,

4.1

and, forn ≥ 1, we have

Yn: yn− yn−1ⁿ

ν0

anνλνaν. 4.2

We may writenoting that vii implies that a_n0  0,

Y_nⁿ

ν1

a_nνλ_ν ν

νa_ν

ⁿ

ν1

anνλν

ν

_ν

r1

rar−^ν−1

r1

rar

ⁿ⁻¹

ν1

Δ_ν

a_nνλ_ν ν

_ν

r1

ra_r a_nnλ_n n

n r1

ra_r

ⁿ⁻¹

ν1

Δ_νa_nνλ_νν 1 ν t_ν ⁿ⁻¹

ν1

a_{n,ν 1}Δλ_νν 1 ν t_ν

ⁿ⁻¹

ν1

an,ν 1λν 11

νtν n 1annλntn

n

 Tn1 Tn2 Tn3 Tn4, say.

4.3

To complete the proof it is sufficient, by Minkowski’s inequality, to show that

∞ n1

n^{δk k−1}|Tnr|^k< ∞, for r  1, 2, 3, 4. 4.4

From the definition of A and using vi and vii it follows that

an,ν 1≥ 0. 4.5

Using H ¨older’s inequality

I1:^m

n1

n^{δk k−1}|T_n1|^k^m

n1

n^{δk k−1}



n−1

ν1

Δ_νa_nνλ_νν 1 ν t_ν



k

 O1^{m 1}

n1

n^{δk k−1} _n−1



ν1

|Δ_νa_nν||λ_ν||t_ν| _k

 O1^{m 1}

n1

n^{δk k−1} _n−1



ν1

|Δνanν||λν|^k|tν|^k _n−1



ν1

|Δνanν| _k−1

,

Δνanν anν− an,ν 1

 a_nν− a_n−1,ν− a_{n,ν 1} a_{n−1,ν 1}

 a_nν− a_n−1,ν ≤ 0.

4.6

Thus, usingvii,

n−1

ν0

|Δ_νa_nν| ⁿ⁻¹

ν0

a_n−1,ν− a_nν  1 − 1 a_nn a_nn. 4.7

Sinceλn is bounded byLemma 3.3, usingv, ix, xi, i, and condition 3.7 ofLemma 3.4

I1 O1^{m 1}

n1

n^δkna_nn^k−1ⁿ⁻¹

ν1

|λ_ν|^k|t_ν|^k|Δ_νa_nν|

 O1^{m 1}

n1

n^δk _n−1

ν1

|λν|^k−1|λν||Δνanν||tν|^k

 O1^m

ν1

|λ_ν||t_ν|^{km 1}

nν 1

n^δk|Δ_νa_nν|

 O1^m

ν1

ν^δk|λν|aνν|tν|^k

 O1^m

ν1

ν^δk−1|λν||tν|^k

 O1

_m−1



ν1Δ|λν|^ν

r1

r^δk−1|tr|^k |λm|^m

r1

r^δk−1|tr|^k 

 O1^m−1

ν1

|Δλ_ν|X_ν O1|λm|X_m

 O1^m

ν1

β_νX_ν O1|λm|Xm

 O1.

4.8

Using H ¨older’s inequality,

I2:^{m 1}

n2

n^{δk k−1}|T_n2|^k^{m 1}

n2

n^{δk k−1}



n−1

ν1

a_{n,ν 1}Δλ_νν 1 ν t_ν



k

 O1^{m 1}

n2

n^{δk k−1} _n−1



ν1

a_{n,ν 1}|Δλ_ν||t_ν| _k

 O1^{m 1}

n2

n^{δk k−1} _n−1



ν1

|Δλν||tν|^ka_{n,ν 1 n−1}



ν1

a_{n,ν 1}|Δλν| _k−1

.

4.9

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

∞ n1

|Δλn| ≤ 2^∞

n1

∞ kn

|Δ|Δλk||  2^∞

k1

|Δ|Δλk|| 4.10

holds. Thus byLemma 3.3,3.4 implies that_∞

n1|Δλn| converges. Therefore, there exists a positive constantM such that_∞

n1|Δλ_n| ≤ M and from the properties of matrix A, we obtain

n−1

ν1

a_{n,ν 1}|Δλ_k| ≤ Ma_nn. 4.11

We have, usingv and x,

I2 O1^{m 1}

n2

n^δknann^k−1n−1

ν1

a_{n,ν 1}β_ν|tν|^k

 O1^m

ν1

β_ν|t_ν|^{km 1}

nν 1

n^δka_{n,ν 1}.

4.12

Therefore,

I2 O1^m

ν1

ν^δkβν|tν|^k

 O1^m

ν1

νβν|tν|^k ν ν^δk.

4.13

Using summation by parts,2.2, xi, and condition 3.6 and 3.7 ofLemma 3.4

I2: O1^m−1

ν1

Δ νβν^ν

r1

r^δk−1|tr|^k O1mβm

m r1

r^δk−1|tr|^k

 O1^m−1

ν1

νΔβνXν O1^m−1

ν1

β_{ν 1}X_{ν 1} O1

 O1.

4.14

Using H ¨older’s inequality andviii,

m 1

n2

n^k−1|T_n3|^k^{m 1}

n2

n^{δk k−1}



n−1

ν1

a_{n,ν 1}λ_{ν 1}1 νt_ν



k

≤^{m 1}

n2

n^{δk k−1} _n−1



ν1

|λν 1|an,ν 1

ν |tν| _k

 O1^{m 1}

n2

n^{δk k−1} _n−1

ν1

|λν 1|an,ν 1|tν|aνν

_k

 O1^{m 1}

n2

n^{δk k−1} _n−1

ν1

|λν 1|^kaνν|tν|^kan,ν 1

_n−1

ν1

aνν|an,ν 1| _k−1

.

4.15

Using boundedness of{λ_n}, v, x, xi, Lemmas3.3and3.4

I3 O1^{m 1}

n2

n^δkna_nn^k−1ⁿ⁻¹

ν1

|λ_{ν 1}|^ka_νν|t_ν|^ka_{n,ν 1}

 O1^m

ν1

|λ_{ν 1}|a_νν|t_ν|^{km 1}

nν 1

n^δka_{n,ν 1}

 O1^m

ν1

|λ_{ν 1}|ν^δka_νν|t_ν|^k

 O1^m

v1

|λv 1|vavvv^δk−1|tv|^k

 O1^m

v1

|λv 1|v^δk−1|tv|^k.

4.16

Using summation by parts

I3 O1^m−1

v1

|Δλv 1|^v

r1

r^δk−1|tr|^k O1|λm 1|^m

v1

v^δk−1|tv|^k

 O1^m−1

v1

|Δλv 1|^{v 1}

r1

r^δk−1|tr|^k O1|λm 1|^{m 1}

v1

v^δk−1|tv|^k

 O1^m−1

v1

|Δλ_{v 1}|X_{v 1} O1|λm 1|X_{m 1}

 O1^m−1

v1

βv 1Xv 1 O1|λm 1|Xm 1

 O1.

4.17

Finally, using boundedness of{λ_n}, and v we have

m n1

n^{δk k−1}|Tn4|^k^m

n1

n^{δk k−1}

n 1annλntn

n

^k

 O1^m

n1

n^δkann|λn||tn|^k

 O1,

4.18

as in the proof ofI1.

5. Corollaries and Applications to Weighted Means

Setting δ  0 in Theorem 2.1 and Theorem 2.2 yields the following two corollaries, respectively.

Corollary 5.1. Let A satisfy conditions (v)–(viii) and let {βn} and {λn} be sequences satisfying conditions (i), (ii), and 2.1. If {Xn} is a quasi f-increasing sequence, where {fn} :

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

m n1

1

n|t_n|^k OXm, m −→ ∞, 5.1

are satisfied then the series

anλnis summable|A|_k, k ≥ 1.

Proof. If we takeδ  0 inTheorem 2.1then conditionxi reduces condition 5.1.

Corollary 5.2. Let A satisfy conditions (v)–(viii) and let {βn} and {λn} be sequences satisfying conditions (i), (ii), and2.1. If {Xn} is a quasi β-power increasing sequence for some 0 ≤ β < 1 and conditions2.3 and 5.1 are satisfied then the seriesa_nλ_nis summable|A|_k,k ≥ 1.

Corollary 5.3. Let {pn} be a positive sequence such that Pn:_n

i0pi → ∞, as n → ∞ satisfies

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 (i), (ii), 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 (xi) and 2.2 are satisfied then the series,a_nλ_nis summable|N, p_n, δ|_kfork ≥ 1.

Proof. In Theorem 2.1, set A  N, p_n. Conditions i and ii of Corollary 5.3 are, respectively, conditionsi and ii ofTheorem 2.1. Conditionv becomes condition 5.2 and conditionsix and x become condition 5.3 for weighted mean method. Conditions vi,

vii, and viii ofTheorem 2.1are automatically satisfied for any weighted mean method.

The following Corollary is the special case ofCorollary 5.3forμ  0.

Corollary 5.4. Let {pn} be a positive sequence satisfying 5.2, 5.3 and let {Xn} be a quasi β-power increasing sequence for some 0≤ β < 1. Then under conditions (i), (ii), (xi), 2.1, and 2.3,

anλn

is summable|N, pn, δ|_k, k ≥ 1.

References

1 B. E. Rhoades and E. Savas¸, “A summability factor theorem for generalized absolute summability,”

Real Analysis Exchange, vol. 31, no. 2, pp. 355–363, 2006.

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 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 E. Savas¸, “A note on generalized |A|_k-summability factors for infinite series,” Journal of Inequalities and Applications, vol. 2010, Article ID 814974, 10 pages, 2010.

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 L. Leindler, “A note on the absolute Riesz summability factors,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, no. 4, article 96, 2005.

9 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.