Journal of Informatics and Mathematical Sciences
Volume 4 (2012), Number 2, pp. 241–248
© RGN Publications
http://www.rgnpublications.com
On the Absolute Summability Factors of Infinite Series
involving Quasi-f-power Increasing Sequence
W.T. Sulaiman
Abstract. In this note we improve a result concerning absolute summability
factor of an infinite series via quasi β-power increasing sequence achieved by
Sevli and Leindler [1].
1. Introduction
A positive sequence (bn ) is said to be almost increasing if exist a positive
increasing sequence (cn ) and two positive constants A and B such that Acn ≤ bn ≤
Bcn .
A positive sequence a = (an ) is said to be quasi β-power increasing if there
exists a constant K = k(β, a) ≥ 1 such that
K nβ an ≥ mβ am
(1.0)
holds for all n ≥ m. If (1.0) stays with β = 0 then a is called a quasi increasing
sequence. It should be noted that every almost increasing sequence is a quasi
β-power increasing sequence for any nonnegative β, but the converse need not
be true as can be seen by taking an = n−β .
A positive sequence α = (αn ) is said to be a quasi-f-power increasing sequence,
f = ( f n ), if there exits a constant K = K(α, f ) such that
K f n αn ≥ f m αm
holds for n ≥ m ≥ 1 (see [3]). Clearly if α is quasi-f-power increasing sequence,
then α f is quasi increasing sequence.
Let T be a lower triangular matrix, (sn ) a sequence, and
n
X
Tn :=
t nv s v .
v=0
2010 Mathematics Subject Classification. 40A05, 40D15, 40F05.
Key words and phrases. Infinite series; Absolute summability; Summability factor.
(1.1)
242
W.T. Sulaiman
A series
P
an is said to be summable |T |k , k ≥ 1 if
∞
X
nk−1 |∆Tn−1 |k < ∞.
(1.2)
n=1
b , with
Given any lower triangular matrix T one can associate the matrices T̄ and T
entries defined by
n
X
n, i = 0, 1, 2 . . . , bt nv = t̄ nv − t̄ n−1,v
t ni ,
t̄ nv =
t=v
respectively. With sn =
i=0
n
X
n
X
t nv s v =
tn =
n
P
v=0
ai λi ,
v
X
t nv
v=0
n
X
ai λi =
i=0
i=0
n
X
Yn := t n − t n−1 =
n
X
ai λi
n
X
t nv =
v=i
n−1
X
t̄ ni ai λi −
t=0
n
X
t̄ n−1,i ai λi =
i=0
t̄ ni ai λi ,
(1.3)
i=0
bt ni ai λi ,
as t̄ n−1,n = 0. (1.4)
i=0
We call T a triangle if T is lower triangular and t nn 6= 0 for all n. We designate
A = (anv ) to be of class Ω if the following holds
(i) is lower triangular
(ii) anv ≥ 0, n, v = 0, 1, . . .
(iii) an−1,v ≥ anv , for n ≥ v + 1
(iv) an0 = 1, n = 0, 1, . . .
By t n we denote the nth (C, 1) mean of the sequence (nan ), that is t n =
Very recently, Selvi and Leindler [1] proved the following result
1
n+1
n
P
v=1
va v .
Theorem 1.1. Let A ∈ Ω satisfying
nann = O(1),
n→∞
(1.5)
and let (λn ) be a sequence of real numbers satisfying
m
X
λn = o(m),
m→∞
(1.6)
n=1
and
m
X
|∆λn | = o(m),
m → ∞.
(1.7)
n=1
If (X n ) is a quasi-f-increasing sequence and the conditions
m
X
1
n=1
n
|t n |k = O(X m ),
m → ∞,
(1.8)
On the Absolute Summability Factors of Infinite Series involving Quasi-f-power Increasing Sequence
243
∞
X
nX n (β, µ)|∆|∆λn | | < ∞,
(1.9)
n=1
P
are satisfied then the series an λn is summable |A|k ≥ 1, where ( f n ) = (nβ logµ n),
µ ≥ 0, 0 ≤ β < 1 and X n (β, µ) = max{nβ (log n)µ , log n}.
2. Main Result
The purpose of this paper is to give the following
Theorem 2.1. Let A ∈ Ω, Let (X n ) be a quasi-f-power increasing sequence and let
(λn ) be sequence of real numbers all are satisfying (1.5) and
λn = o(1),
n → ∞,
|λn |X n = O(1),
(2.1)
n → ∞,
(2.2)
∞
X
(2.3)
nX n |∆|∆λn | | < ∞,
n=1
m
X
|t n |k
n=1
nX nK−1
(2.4)
= O(X m ),
n−1
X
bn,v+1 a vv = O(ann ),
a
n → ∞,
(2.5)
v=1
then the series
P
Q
Pµ
N
an λn is summable |A|k , k ≥1, where f =( f n ), f n = nβ
logµ n ,
µ=1
0 ≤ β < 1, 1 ≤ N < ∞, Pµ > 0, and logµ n = log(logµ−1 n).
The goodness and advantage of Theorem 2.1 follows from the following remark.
Remark. (1) Although the condition (2.5) is added (in comparing with
Theorem 1.1) but this condition is trivial. As an example if we are putting
anv = p v /Pn (in order to have the |N̄ , pn |k summability), (2.5) is obvious by
the following
n−1
X
bn,v+1 a vv =
a
v=1
n−1
X
pn Pv p v
v=1
Pn Pn−1 Pv
=
pn
Pn Pn−1
n−1
X
Pv =
v=1
pn
Pn
= ann = O(ann ).
(2) Let us define the following two groups of conditions
group I = {(1.6), (1.7), (1.9)},
group J = {(2.1), (2.2), (2.3)}.
It is clear that (see [1]) I ⇒ J , but not the converse. Therefore the conditions
of Theorem 3.1 are weaker than those of Theorem 1.1.
244
W.T. Sulaiman
(3) The condition (1.8) in using impose us to loose powers of estimation. For
example in the proof of Theorem 1.1, through the estimations of I1 and I2 ,
the power λn |k−1 has been lost through the estimation when it has been
substituted by (|λn |k−1 = O(1)), while we have not this case on using (2.4).
(4) The sequence (X n ) used in Theorem 2.1 is more general and in some sense
make some condition is weaker than (X n ) defined in Theorem 1.1.
3. Lemmas
Lemma 3.1. The conditions (1.6), (1.7) and (1.9) implies λn = O(1), as n → ∞.
For the proof see [1].
Lemma 3.2. Let A ∈ Ω. Then
n−1
X
|∆n an−1m,v | ≤ ann ,
v=1
m+1
X
|∆n an−1m,v | ≤ a vv .
n=v+1
Lemma 3.3. Let A ∈ Ω. Then
bn+1,v ≤ ann
a
for n ≥ v + 1,
m+1
X
bn+1,v ≤ 1,
a
v = 0, 1, . . . .
n=v+1
Lemma 3.4. Let (X n ) be as defined in Theorem 2.1. Then conditions (2.1) and (2.3)
implies
nX n |∆λn | = O(1),
as n → ∞,
(3.1)
∞
X
X |∆λn | < ∞.
(3.2)
n=1
Proof. Let f n be as defined in Theorem 2.1. Then by (2.1), ∆|∆λn | → 0, and hence
∞
X
nX n |∆λn | = nX n ∆|∆λ v |
v=n
∞
X
≤ nX n
|∆|∆λ v | | = O(1)n f n−1 f n X n
= O(1)n f n−1
∞
X
f v X v |∆|∆λ v | |
v=n
∞
X
v=n
|∆|∆λ v | |
v=n
v=n
= O(1)
∞
X
v f v−1 f v X v |∆|∆λ v | |
On the Absolute Summability Factors of Infinite Series involving Quasi-f-power Increasing Sequence
245
∞
X
= O(1)
vX v |∆|∆λ v | |
v=n
= O(1),
∞
X
∞
X
X n |∆λn | ≤
n=1
∞
X
∞
X
|∆λ v | | =
Xn
v=n
n=1
∞
X
= O(1)
f n−1
|∆|∆λ v | |
v=n
n=1
∞
X
f v X v |∆|∆λ v | |
v=n
n=1
∞
X
= O(1)
∞
X
f n−1 f n X n
v
X
f v X v |∆|∆λ v | |
v=1
f n−1 .
n=1
Now, as
v
X
f n−1 =
n=1
v
X
nβ+ε f n−1 n−β−ε ,
0<β +ε<1
n=1
≤v
β+ε
f v−1
Z
v
X
−β−ε
n
= O(1)v
β+ε
f v−1
v
x −β−ε d x = O(1)v f v−1 .
0
n=1
Then, we have
∞
∞
X
X
X n |∆λn | = O(1)
nX n |∆|∆λn | | < ∞ .
n=1
¤
n=1
4. Proof of Theorem 1.2
Let yn denotes the nth term of the A-transform of the series
(1.4), we have, for n ≥ 1
P
an λn . Then by
Yn = yn − yn−1
n
X
bnv a v λ v
a
=
v=0
v
n−1 n
X
bnv λ v X
a
ann λn X
=
∆
ra r +
va v
v
n v=1
v=1
r=1
=
(n + 1)ann t n λn
n
n−1
X
−
∆n an−1,v λ v t v
v=1
v+1
v
n−1
X
bn,v+1 λ v t v
a
+
v=1
1
v
n−1
X
bn,v+1 ∆λ v t v
a
+
v=1
= Yn1 + Yn2 + Yn3 + Y4 .
To complete the proof, by Minkowski’s inequality, it is sufficient to show that
∞
X
nk−1 |Ynv |k < ∞, v = 1, 2, 3, 4.
n−1
246
W.T. Sulaiman
Applying Holder’s inequality,
m+1
X
n
k−1
m+1
X
k
|Yn1 | =
n=1
n
(n + 1)ann t n λn k
n
k−1 n=1
m+1
X
= O(1)
|t n |k λn |(|λn |X n )k−1
(nan n)k−1
nX nk−1
n=1
= O(1)
m
X
|λ v | |t v |k
vX vk−1
v=1
m−1
X
= O(1)
∆|λ v |
v=1
v
X
|t r |k
rX rk−1
r=0
+ O(1)|λm |
m
X
|t v |k
v=1
vX vk−1
m
X
= O(1)
|∆λ v |X v + O(1)|λm |X m
v=1
= O(1),
m+1
X
n
n=1
k−1
k
m+1
X
|Yn2 | =
n
k
n−1
X
v + 1
∆n an−1,v λ v
v k−1 n=1
v=1
m+1
X
= O(1)
n
n=1
m+1
X
= O(1)
v=1
nk−1
m+1
X
n=1
m+1
X
= O(1)
|∆n an−1,v | |λ v | |t V |
nk−1
n−1
X
|∆n an−1,v | |λ v |k |t V |k
v=1
(nann )k−1
n=1
m
X
= O(1)
|λ v |k |t v |k
m
X
n−1
X
m+1
X
|∆n , an−1,v |
n=v+1
a vv |λ v |k |t v |k
m
X
|λ v | |t v |k
v=1
= O(1),
vX vk−1
as in the case of Yn1 .
k−1
∆n an−1,v |
|∆n an−1,v | |λ v |k |t v |k
v=1
= O(1)
X
n−1
v=1
v=1
v=1
= O(1)
k
X
n−1
v=1
n=1
= O(1)
n−1
k
X
v + 1
∆n an−1,v , λ v t v
v k−1 On the Absolute Summability Factors of Infinite Series involving Quasi-f-power Increasing Sequence
m+1
X
nk−1 |Yn3 |k =
n=1
n−1
k
X
1
bn,v+1 λ v t v nk−1 a
v
v=1
n=1
m+1
X
m+1
X
= O(1)
nk−1
X
n−1
n=1
= O(1)
n
k+1
n=1
m+1
X
= O(1)
k
a v v |b
an,v+1 | |λ v | |t v |
n
n=1
k
m+1
X
|Yn4 | =
n
bn,v+1
a vv a
n−1
X
bn,v+1 |λ v |k |t v |k
a vv a
v=1
vX Vk−1
bn,v+1 = O(1)
a
n=v+1
m
X
|λ v | |t v |k
v=1
V x vk−1
n−1
k
X
bn,v+1 ∆λ v t v a
k−1 v=1
m+1
X
= O(1)
n
k−1
bn,v+1 |∆λ v | |t v |
a
v=1
m+1
X
= O(1)
n
n=1
m+1
X
= O(1)
k−1
n−1
X
(b
an,v+1 )
(nann )k−1
bn,v+1
a
X vk−1
vX vk−1
X v |∆λ v |
v=1
|t v |k
|t v |k
v
X
|t r |k
r=1
r X rk−1
+ O(1)m|∆λm |X m
m
X
|∆λ v |X v + O(1)
= O(1).
X vk−1
k−1
bn,v+1
a
X
v=1
|∆λ v |
X
n−1
n=v+1
∆(v|∆λ v |)
= O(1)
|t v |
k
m+1
X
m−1
X
v=1
X vk−1
n−1
X
|t v |k
m
X
v|∆λ v |
= O(1)
|∆λ v |
v=1
m
X
|∆λ v |
v=1
k
v=1
n=1
v=1
k
X
n−1
n=1
= O(1)
k−1
as in the case of Yn1 .
n=1
= O(1)
X
n−1
v=1
(nann )k−1
v=1
k−1
k
v=1
m
X
X
|λ v | |t v |K m+1
= O(1),
k
v
n−1
X
n=1
= O(1)
bn,v+1 |λ v | |t v |
a
v=1
m+1
X
m+1
X
247
v|∆|∆λ v | |X v + O(1)m|∆λm |X m
v=1
248
W.T. Sulaiman
References
[1] H. Sevli and L. Leindler, On the absolute summability factors of infinite series involving
quasi-power-increasing sequences, Computer and Mathematics with Applications 57
(2009), 702–709.
[2] B.E. Rhoades, Inclusion theorems for absolute matrix summability methods, J. Math.
Anal. Appl. 238 (1999), 82–90.
[3] W.T. Sulaiman, Extension on absolute summability factors of infinite series, J. Math.
Anal. Appl. 322 (2006), 1224–1230.
W.T. Sulaiman, Department of Computer Engineering, College of Engineering,
University of Mosul, Iraq.
E-mail: [email protected]
Received
Accepted
March 8, 2011
May 5, 2012