Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 4, Article 135, 2006
A NOTE ON |N̄ , pn |k SUMMABILITY FACTORS
S.M. MAZHAR
D EPARTMENT OF M ATHEMATICS AND C OMPUTER S CIENCE
K UWAIT U NIVERSITY
P.O. B OX N O . 5969, K UWAIT - 13060.
sm_mazhar@hotmail.com
Received 18 March, 2006; accepted 03 July, 2006
Communicated by H. Bor
A BSTRACT. In this note we investigate the relation between two theorems proved by Bor [2, 3]
on |N̄ , pn |k summability of an infinite series.
Key words and phrases: Absolute summability factors.
2000 Mathematics Subject Classification. 40D15, 40F05, 40G05.
1. I NTRODUCTION
P
Let
an be a given infinite series with {sn } as the sequence of its n-th partial sums. Let
{pn } be a sequence of positive constants such that Pn = p0 + p1 + p2 + · · · + pn −→ ∞ as
n −→ ∞.
Let
n
1 X
tn =
p ν sν .
Pn ν=1
P
P
The series
an is said to be summable |N̄ , pn | if ∞
1 |tn − tn−1 | < ∞. It is said to be
summable |N̄ , pn |k , k ≥ 1 [1] if
k−1
∞ X
Pn
(1.1)
|tn − tn−1 |k < ∞,
p
n
1
and bounded [N̄ , pn ]k , k ≥ 1 if
(1.2)
n
X
pν |sν |k = O(Pn ),
n −→ ∞.
1
Concerning |N̄ , pn | summability factors of
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
081-06
P
an , T. Singh [6] proved the following theorem:
2
S.M. M AZHAR
Theorem A. If the sequences {pn } and {λn } satisfy the conditions
∞
X
(1.3)
pn |λn | < ∞,
1
(1.4)
C is a constant, and if
P
Pn |∆λn | ≤ Cpn |λn |,
P
an is bounded [N̄ , pn ]1 , then an Pn λn is summable |N̄ , pn |.
Earlier in 1968 N. Singh [5] had obtained the following theorem.
P
Theorem B. If an is bounded [N̄ , pn ]1 and {λn } is a sequence satisfying the following conditions
∞
X
pn |λn |
< ∞,
(1.5)
P
n
1
Pn
∆λn = O(|λn |),
pn
(1.6)
then
P
an λn is summable |N̄ , pn |.
In order to extend these theorems to the summability |N̄ , pn |k , k ≥ 1, Bor [2, 3] proved the
following theorems.
P
Theorem C. Under the conditions (1.2), (1.3) and (1.4), the series
an Pn λn is summable
|N̄ , pn |k , k ≥ 1.
P
Theorem D. If an is bounded
[N̄ , pn ]k , k ≥ 1 and {λn }, is a sequence satisfying the condiP
tions (1.4) and (1.5), then an λn is summable |N̄ , pn |k .
2. R ESULTS
In this note we propose to examine the relation between Theorem C and Theorem D.
We recall that recently Sarigol and Ozturk [4] constructed an example to demonstrate
that the
P
hypotheses of Theorem A are not sufficient for the summability |N̄ , pn | of
an Pn λn . They
proved that Theorem A holds true if we assume the additional condition
pn+1 = O(pn ).
(2.1)
From (1.4) we find that
∆λn Cpn
λ
n+1
= 1 −
≤
,
λn λn Pn
Hence
λn+1 λn+1
=
−
1
+
1
λn λn
λn+1 ≤ 1 −
+1
λn Cpn
≤
+ 1 ≤ C.
Pn
Thus |λn+1 | ≤ C|λn |, and combining this with (2.1) we get
(2.2)
pn+1 |λn+1 | ≤ Cpn |λn |.
J. Inequal. Pure and Appl. Math., 7(4) Art. 135, 2006
http://jipam.vu.edu.au/
|N̄ , pn |k S UMMABILITY FACTORS
3
Clearly (2.1) and (1.4) imply (2.2). However (2.2) need not imply (2.1) or (1.4). In view of
∆(Pn λn ) = Pn ∆λn − pn+1 λn+1
it is clear that if (2.2) holds, then the condition (1.4) is equivalent to the condition
|∆(Pn λn )| ≤ Cpn |λn |.
(2.3)
It can be easily verified that a corrected version of Theorem A and Theorem C and also a slight
generalization of the result of Sarigol and Ozturk for k = 1 can be stated as
P
Theorem 2.1. Under the conditions (1.2), (1.3) (2.2) and (2.3) the series an Pn λn is summable |N̄ , pn |k , k ≥ 1
We now proceed to show that Theorem 2.1 holds good without condition (2.2).
Thus we have:
P
Theorem 2.2. Under the conditions (1.2), (1.3) and (2.3) the series
an Pn λn is summable
|N̄ , pn |k , k ≥ 1.
To prove Theorem 2.2 we first prove the following lemma.
Lemma 2.3. Under the conditions of Theorem 2.2
m
X
(2.4)
pn |λn ||sn |k = O(1) as m −→ ∞.
1
3. P ROOFS
Proof of Lemma 2.3. In view of (1.3) and (2.3)
∞
X
|∆(λn Pn )| ≤ C
1
∞
X
pn |λn | < ∞,
1
so it follows that {Pn λn } ∈ BV and hence Pn |λn | = O(1).
Now
m
X
1
k
pn λn ||sn | =
m−1
X
∆|λn |
1
= O(1)
n
X
k
pν |sν | + |λm |
ν=1
m−1
X
m
X
pν |sν |k
ν=1
|∆λn |Pn + O(|λm |Pm )
1
m−1
X
= O(1)
!
|∆(Pn λn )| + pn+1 |λn+1 |
+ O(1)
1
= O(1)
m−1
X
1
pn |λn | + O(1)
m
X
pn+1 |λn+1 | + O(1)
1
= O(1).
J. Inequal. Pure and Appl. Math., 7(4) Art. 135, 2006
http://jipam.vu.edu.au/
4
S.M. M AZHAR
P
Proof of Theorem 2.2. Let Tn denote the nth N̄ , pn means of the series an Pn λn . Then
Tn =
n
ν
1 X X
pν
ar P r λ r
Pn ν=0 r=0
n
1 X
=
(Pn − Pν−1 ) aν Pν λν .
Pn ν=0
so that for n ≥ 1
Tn − Tn−1 =
n
pn X
Pν−1 aν Pν λν
Pn Pn−1 ν=1
n−1
pn X
=
∆(Pν−1 Pν λν )sν + pn λn sn
Pn Pn−1 ν=1
= L1 + L2 , say.
Thus to prove the theorem it is sufficient to show that
k−1
∞ X
Pn
|Lν |k < ∞,
pn
1
ν = 1, 2.
Now
|∆(Pν−1 Pν λν )| ≤ pν Pν |λν | + Pν |∆(Pν λν )|
≤ Cpν Pν |λν |
in view of (2.3). So
m+1
m+1
X Pn k−1
X pn
|L1 |k = O(1)
pn
P Pk
n=2
n=2 n n−1
= O(1)
m+1
X
n=2
= O(1)
m+1
X
n=2
= O(1)
m
X
pn
k
Pn Pn−1
n−1
X
!k
pν Pν |λν ||sν |
ν=1
n−1
X
!
(Pν |λν |)k |sν |k pν
ν=1
n−1
X
!k−1
pν
ν=1
n−1
pn X
Pν |λν |pν |sν |k
Pn Pn−1 ν=1
pν |λν |sν |k = O(1)
ν=1
in view of the lemma and Pn |λn | = O(1).
Also
m+1
m+1
X Pn k−1
X
k
|L2 | = O(1)
pn |λn |k |sn |k Pnk−1
p
n
1
1
= O(1)
m+1
X
pn |λn ||sn |k
1
= O(1).
This proves Theorem 2.2.
J. Inequal. Pure and Appl. Math., 7(4) Art. 135, 2006
http://jipam.vu.edu.au/
|N̄ , pn |k S UMMABILITY FACTORS
5
Thus a generalization of a corrected version of Theorem C is Theorem 2.2. Writing λn =
µn Pn the conditions (1.5) and (1.4) become
∞
X
(3.1)
pn |µn | < ∞,
1
(3.2)
|∆(Pn µn )| ≤ Cpn |µn |,
consequently
Theorem D can be stated as:
P
an is bounded [N̄ , Pn ]k , k ≥ 1 and {µn } is a sequence satisfying (3.1) and (3.2) then
PIf
an Pn µn is summable |N̄ , pn |k , k ≥ 1.
Thus Theorem D is the same as Theorem 2.2 which is a generalization of the corrected
version of Theorem C.
R EFERENCES
[1] H. BOR, On |N̄ , pn |k summability methods and |N̄ , pn |k summability factors of infinite series, Ph.D.
Thesis (1982), Univ. of Ankara.
[2] H. BOR, On |N̄ , pn |k summability factors, Proc. Amer. Math. Soc., 94 (1985), 419–422.
[3] H. BOR, On |N̄ , pn |k summability factors of infinite series, Tamkang J. Math., 16 (1985), 13–20.
[4] M. ALI SARIGOL AND E. OZTURK, A note on |N̄ , pn |k summability factors of infinite series,
Indian J. Math., 34 (1992), 167–171.
[5] N. SINGH, On |N̄ , pn | summability factors of infinite series, Indian J. Math., 10 (1968), 19–24.
[6] T. SINGH, A note on |N̄ , pn | summability factors of infinite series, J. Math. Sci., 12-13 (1977-78),
25–28.
J. Inequal. Pure and Appl. Math., 7(4) Art. 135, 2006
http://jipam.vu.edu.au/