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¸,[email protected]
Received 23 June 2010; Revised 3 September 2010; Accepted 15 September 2010
Academic Editor: J. Szabados
Copyrightq2010 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, whereAis 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¸2006by using quasif-increasing sequences.
1. Introduction
Recently, Rhoades and Savas¸1obtained sufficient conditions foranλnto 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 bvif n|Δλn| < ∞.Let bv0
bv∩c0,wherec0denotes the set of all null sequences.
LetAbe a lower triangular matrix,{sn}a sequence. Then
An: n
ν0
anνsν. 1.1
A seriesan, with partial sumssn, is said to be summable|A|k, k≥1 if
∞
n1 nk−1|A
n−An−1|k<∞, 1.2
and it is said to be summable|A, δ|k, k≥1 andδ≥0 ifsee,2
∞
n1
nδk k−1|A
A positive sequence{bn}is said to be an almost increasing sequence if there exist an
increasing sequence{cn}and positive constantsAandBsuch thatAcn≤bn≤Bcnsee,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:{γn}is said to be a quasiβ-power increasing sequence if there
exists a constantKKβ, γ≥1 such that
Knβγ
n≥mβγm 1.4
holds for alln ≥ m ≥ 1. It should be noted that every almost increasing sequence is quasi
β-power increasing sequence for any nonnegativeβ, but the converse need not be true as can be seen by taking an example, sayγnn−βforβ >0see,4. A sequence satisfying1.4for
β0 is called a quasi-increasing sequence.It is clear that if{γn}is quasiβ-power increasing
then{nβγn}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 thatKfnγn ≥ fmγm holds for alln ≥ m ≥ 1, where
f:{fn}{nβlognμ}, μ >0,0< β <1,see,5.
We may associate withAtwo lower triangular matricesAandAas follows:
anv n
rvanr, n, v0,1, . . . ,
anvanv−an−1,v, n1,2, . . . ,
1.5
where
a00a00a00. 1.6
Given any sequence{xn}, the notationxn O1means thatxn O1and 1/xn
O1.For any matrix entryanv,Δvanv :anv−an,v 1.
Rhoades and Savas¸1proved 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,
iilimβn0,
iii∞n1n|Δβn|Xn<∞,
iv|λn|Xn O1.
LetAbe a lower triangular matrix with nonnegative entries satisfying
vnannO1,
vian−1,ν≥anνforn≥ν 1,
viiinν1−1aννanν 1 Oann,
ixmn 1ν 1nδk|Δνanν|Oνδkaννand
xmn 1ν 1nδkanν 1Oνδk.
If
ximn1nδk−1|t
n|kOXm, wheretn: 1/n 1nk1kak,
then the seriesanλnis summable|A, δ|k, k≥1.
It should be noted that, if{Xn}is an almost increasing sequence, then conditioniv
implies that the sequence{λn}is bounded. However, if{Xn}is a quasiβ-power increasing sequence or a quasi f-increasing sequence, iv does not imply that λ is bounded. For example, the sequence{Xm} defined byXm m−β is trivially a quasiβ-power increasing
sequence for eachβ >0.Ifλ{mδ},for any 0< δ < β,thenλmXmmδ−βO1,butλis 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,{λn} ∈ 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 {βn} and {λn} be sequences satisfying conditions (i) and (ii) ofTheorem 1.1and
m
n1
λnom, m−→ ∞. 2.1
If{Xn}is a quasif-increasing sequence and condition (xi) and
∞
n1
nXnβ, μΔβn<∞ 2.2
are satisfied then the seriesanλn is summable|A, δ|k,k ≥ 1,where {fn} : {nβlognμ}, μ ≥
0,0≤β <1,andXnβ, μ: nβlognμXn.
The following theorem is the special case ofTheorem 2.1forμ0.
Theorem 2.2. Let A satisfy conditions (v)–(x) and let {βn} and {λn} be sequences satisfying
conditions (i), (ii), and2.1. If{Xn}is a quasiβ-power increasing sequence for some0 ≤ β < 1 and conditions (xi) and
∞
n1
nXnβΔβn<∞ 2.3
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{Xn},{βn}, and{λn}as taken in the statement of theTheorem 2.1, also in the statement ofTheorem 2.2with the special case
μ0,conditions{λn} ∈bv0andivhold.
3. Lemmas
We will need the following lemmas for the proof of our mainTheorem 2.1.
Lemma 3.1see8. Let{ϕn}be a sequence of real numbers and denote
Φn: n
k1
ϕk, Ψn:
∞
kn
Δϕk. 3.1
IfΦnonthen there exists a natural numberNsuch that
ϕn≤2Ψn 3.2
for alln≥N.
Lemma 3.2 see9. If{Xn}is a quasif-increasing sequence, where{fn} {nβlognμ}, μ ≥
0,0≤β <1,then conditions2.1ofTheorem 2.1,
m
n1
|Δλn|om, m−→ ∞, 3.3
∞
n1
nXnβ, μ|Δ|Δλn||<∞, 3.4
whereXnβ, μ nβlognμXn,imply conditions (iv) and
λn−→0, n−→ ∞. 3.5
Lemma 3.3 see7. If{Xn}is a quasif-increasing sequence, where{fn} {nβlognμ}, μ ≥
0,0≤β <1,then under conditions (i), (ii),2.1, and2.2, conditions (iv) and3.5are satisfied.
Lemma 3.4see7. Let{Xn}be a quasif-increasing sequence, where{fn}{nβlognμ},μ≥ 0,0≤β <1.If conditions (i), (ii), and2.2are satisfied, then
nβnXnO1, 3.6
∞
n1
4. Proof of
Theorem 2.1
Proof. Letynbe thenth term of theAtransform of the partial sums ofni0λiai. Then we have
yn: n
i0 anisi
n
i0 ani
i
ν0 λνaν
n
ν0 λνaν
n
iν
ani n
ν0
anνλνaν,
4.1
and, forn≥1, we have
Yn:yn−yn−1
n
ν0
anνλνaν. 4.2
We may writenoting thatviiimplies thatan00,
Yn n
ν1 a
nνλν
ν νaν
n
ν1 a
nνλν
ν
ν
r1 rar−
ν−1
r1 rar
n−1
ν1 Δν
a
nνλν
ν
ν
r1
rar annnλn n
r1 rar
n−1
ν1
Δνanνλννν1tν n−1
ν1
an,ν 1Δλννν1tν
n−1
ν1
an,ν 1λν 11νtν n 1nannλntn
Tn1 Tn2 Tn3 Tn4, say.
4.3
To complete the proof it is sufficient, by Minkowski’s inequality, to show that
∞
n1
nδk k−1|T
nr|k<∞, forr1,2,3,4. 4.4
From the definition ofAand usingviandviiit follows that
Using H ¨older’s inequality
I1:
m
n1
nδk k−1|T
n1|k
m
n1
nδk k−1
n−1
ν1
Δνanνλννν1tν
k
O1
m 1
n1 nδk k−1
n−1
ν1
|Δνanν||λν||tν|
k
O1
m 1
n1 nδk k−1
n−1
ν1
|Δνanν||λν|k|tν|k
n−1
ν1
|Δνanν|
k−1 ,
Δνanνanν−an,ν 1
anν−an−1,ν−an,ν 1 an−1,ν 1 anν−an−1,ν ≤0.
4.6
Thus, usingvii,
n−1
ν0
|Δνanν| n−1
ν0
an−1,ν−anν 1−1 annann. 4.7
Sinceλnis bounded byLemma 3.3, usingv,ix,xi,i, and condition3.7ofLemma 3.4
I1O1
m 1
n1 nδkna
nnk−1 n−1
ν1
|λν|k|tν|k|Δνanν|
O1
m 1
n1 nδk
n−1
ν1
|λν|k−1|λν||Δνanν||tν|k
O1
m
ν1
|λν||tν|k m 1
nν 1 nδk|Δ
νanν|
O1
m
ν1 νδk|λ
ν|aνν|tν|k
O1
m
ν1 νδk−1|λ
ν||tν|k
O1
m−1
ν1Δ
|λν| ν
r1 rδk−1|t
r|k |λm| m
r1 rδk−1|t
r|k
O1
m−1
ν1
|Δλν|Xν O1|λm|Xm
O1
m
ν1
βνXν O1|λm|Xm
O1.
Using H ¨older’s inequality,
I2:
m 1
n2
nδk k−1|T
n2|k
m 1
n2
nδk k−1
n−1
ν1
an,ν 1Δλννν1tν
k
O1
m 1
n2 nδk k−1
n−1
ν1
an,ν 1|Δλν||tν|
k
O1
m 1
n2 nδk k−1
n−1
ν1
|Δλν||tν|kan,ν 1 n−1
ν1
an,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 3.3,3.4implies that∞n1|Δλn|converges. Therefore, there exists a
positive constantMsuch that∞n1|Δλn| ≤Mand from the properties of matrixA, we obtain
n−1
ν1
an,ν 1|Δλk| ≤Mann. 4.11
We have, usingvandx,
I2O1
m 1
n2 nδkna
nnk−1 n−1
ν1
an,ν 1βν|tν|k
O1
m
ν1 βν|tν|k
m 1
nν 1 nδka
n,ν 1.
4.12
Therefore,
I2O1
m
ν1 νδkβ
ν|tν|k
O1
m
ν1 νβν|tν|
k
ν νδk.
Using summation by parts,2.2,xi, and condition3.6and3.7ofLemma 3.4
I2:O1
m−1
ν1 Δνβν
ν
r1 rδk−1|t
r|k O1mβm m
r1 rδk−1|t
r|k
O1
m−1
ν1
νΔβνXν O1 m−1
ν1
βν 1Xν 1 O1
O1.
4.14
Using H ¨older’s inequality andviii,
m 1
n2 nk−1|T
n3|k
m 1
n2
nδk k−1
n−1
ν1
an,ν 1λν 1ν1tν
k
≤m 1
n2 nδk k−1
n−1
ν1
|λν 1|an,νν 1|tν|
k
O1
m 1
n2 nδk k−1
n−1
ν1
|λν 1|an,ν 1|tν|aνν
k
O1
m 1
n2 nδk 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{λn},v,x,xi, Lemmas3.3and3.4
I3O1
m 1
n2 nδkna
nnk−1 n−1
ν1
|λν 1|kaνν|tν|kan,ν 1
O1
m
ν1
|λν 1|aνν|tν|k m 1
nν 1 nδka
n,ν 1
O1
m
ν1
|λν 1|νδkaνν|tν|k
O1
m
v1
|λv 1|vavvvδk−1|tv|k
O1
m
v1
|λv 1|vδk−1|tv|k.
Using summation by parts
I3O1
m−1
v1
|Δλv 1|
v
r1 rδk−1|t
r|k O1|λm 1|
m
v1 vδk−1|t
v|k
O1
m−1
v1
|Δλv 1|
v 1
r1 rδk−1|t
r|k O1|λm 1|
m 1
v1 vδk−1|t
v|k
O1
m−1
v1
|Δλv 1|Xv 1 O1|λm 1|Xm 1
O1
m−1
v1
βv 1Xv 1 O1|λm 1|Xm 1
O1.
4.17
Finally, using boundedness of{λn}, andvwe have
m
n1
nδk k−1|T
n4|k
m
n1
nδk k−1n 1annλntn n
k
O1
m
n1 nδka
nn|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} :
{nβlognμ}, μ≥0,0≤β <1,and conditions2.2and
m
n1
1
n|tn|kOXm, m−→ ∞, 5.1
are satisfied then the seriesanλnis summable|A|k, k≥1.
Proof. If we takeδ0 inTheorem 2.1then conditionxireduces condition5.1.
Corollary 5.2. Let Asatisfy conditions (v)–(viii) and let {βn} and {λn} be sequences satisfying
conditions (i), (ii), and2.1. If{Xn}is a quasiβ-power increasing sequence for some0≤β <1and
Corollary 5.3. Let{pn}be a positive sequence such thatPn:ni0pi → ∞,asn → ∞satisfies
npnOPn, asn−→ ∞, 5.2
m 1
nv 1
nδk pn
PnPn−1 O
vδk
Pv
5.3
and let {βn} and{λn}be sequences satisfying conditions (i), (ii), and2.1. If{Xn} is a quasif -increasing sequence, where{fn}:{nβlognμ},μ≥0,0≤β <1,and conditions (xi) and2.2are satisfied then the series,anλnis summable|N, pn, δ|kfork≥1.
Proof. In Theorem 2.1, set A N, pn. Conditions i and ii of Corollary 5.3 are,
respectively, conditionsiandiiofTheorem 2.1. Conditionvbecomes condition5.2and conditionsixandxbecome condition5.3for weighted mean method. Conditionsvi,
vii, andviiiofTheorem 2.1are automatically satisfied for any weighted mean method.
The following Corollary is the special case ofCorollary 5.3forμ0.
Corollary 5.4. Let{pn}be a positive sequence satisfying5.2,5.3and let{Xn}be a quasiβ-power
increasing sequence for some0≤β <1.Then under conditions (i), (ii), (xi),2.1, and2.3,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.
