http://jipam.vu.edu.au/
Volume 7, Issue 4, Article 135, 2006
A NOTE ON|N , p¯ n|k SUMMABILITY FACTORS
S.M. MAZHAR
DEPARTMENT OFMATHEMATICS ANDCOMPUTERSCIENCE
KUWAITUNIVERSITY
P.O. BOXNO. 5969, KUWAIT- 13060.
sm_mazhar@hotmail.com
Received 18 March, 2006; accepted 03 July, 2006 Communicated by H. Bor
ABSTRACT. In this note we investigate the relation between two theorems proved by Bor [2, 3]
on|N , p¯ n|ksummability of an infinite series.
Key words and phrases: Absolute summability factors.
2000 Mathematics Subject Classification. 40D15, 40F05, 40G05.
1. INTRODUCTION
Let P
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 thatPn = p0 +p1 +p2 +· · ·+pn −→ ∞as n−→ ∞.
Let
tn= 1 Pn
n
X
ν=1
pνsν.
The series P
an is said to be summable |N , p¯ n| if P∞
1 |tn− tn−1| < ∞. It is said to be summable|N , p¯ n|k, k ≥1[1] if
(1.1)
∞
X
1
Pn pn
k−1
|tn−tn−1|k<∞,
and bounded[ ¯N , pn]k, k ≥1if (1.2)
n
X
1
pν|sν|k =O(Pn), n −→ ∞.
Concerning|N , p¯ n|summability factors ofP
an,T. Singh [6] proved the following theorem:
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
081-06
Theorem A. If the sequences{pn}and{λn}satisfy the conditions (1.3)
∞
X
1
pn|λn|<∞,
(1.4) Pn|∆λn| ≤Cpn|λn|,
Cis a constant, and ifP
anis bounded[ ¯N , pn]1, thenP
anPnλnis summable|N , p¯ n|.
Earlier in 1968 N. Singh [5] had obtained the following theorem.
Theorem B. IfP
anis bounded[ ¯N , pn]1 and{λn}is a sequence satisfying the following con- ditions
(1.5)
∞
X
1
pn|λn| Pn <∞,
(1.6) Pn
pn∆λn=O(|λn|), thenP
anλnis summable|N , p¯ n|.
In order to extend these theorems to the summability|N , p¯ n|k, k ≥1,Bor [2, 3] proved the following theorems.
Theorem C. Under the conditions (1.2), (1.3) and (1.4), the series P
anPnλn is summable
|N , p¯ n|k, k ≥1.
Theorem D. IfP
anis bounded[ ¯N , pn]k, k ≥1and{λn}, is a sequence satisfying the condi- tions (1.4) and (1.5), thenP
anλnis summable|N , p¯ n|k. 2. RESULTS
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 hypotheses of Theorem A are not sufficient for the summability |N , p¯ n| of P
anPnλn. They proved that Theorem A holds true if we assume the additional condition
(2.1) pn+1 =O(pn).
From (1.4) we find that
∆λn λn
=
1− λn+1 λn
≤ Cpn Pn , Hence
λn+1 λn
=
λn+1
λn −1 + 1
≤
1− λn+1 λn
+ 1
≤ Cpn
Pn + 1 ≤C.
Thus|λn+1| ≤C|λn|, and combining this with (2.1) we get
Clearly (2.1) and (1.4) imply (2.2). However (2.2) need not imply (2.1) or (1.4). In view of
∆(Pnλn) = Pn∆λn−pn+1λn+1
it is clear that if (2.2) holds, then the condition (1.4) is equivalent to the condition
(2.3) |∆(Pnλn)| ≤Cpn|λn|.
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 fork = 1can be stated as
Theorem 2.1. Under the conditions (1.2), (1.3) (2.2) and (2.3) the seriesP
anPnλnis summa- ble|N , p¯ n|k, k ≥1
We now proceed to show that Theorem 2.1 holds good without condition (2.2).
Thus we have:
Theorem 2.2. Under the conditions (1.2), (1.3) and (2.3) the series P
anPnλn is summable
|N , p¯ n|k, k ≥1.
To prove Theorem 2.2 we first prove the following lemma.
Lemma 2.3. Under the conditions of Theorem 2.2 (2.4)
m
X
1
pn|λn||sn|k =O(1)asm−→ ∞.
3. PROOFS
Proof of Lemma 2.3. In view of (1.3) and (2.3)
∞
X
1
|∆(λnPn)| ≤C
∞
X
1
pn|λn|<∞,
so it follows that{Pnλn} ∈BV and hencePn|λn|=O(1).
Now
m
X
1
pnλn||sn|k=
m−1
X
1
∆|λn|
n
X
ν=1
pν|sν|k+|λm|
m
X
ν=1
pν|sν|k
=O(1)
m−1
X
1
|∆λn|Pn+O(|λm|Pm)
=O(1)
m−1
X
1
|∆(Pnλn)|+pn+1|λn+1|
!
+O(1)
=O(1)
m−1
X
1
pn|λn|+O(1)
m
X
1
pn+1|λn+1|+O(1)
=O(1).
Proof of Theorem 2.2. LetTndenote thenth N , p¯ n
means of the seriesP
anPnλn.Then Tn = 1
Pn
n
X
ν=0
pν
ν
X
r=0
arPrλr
= 1 Pn
n
X
ν=0
(Pn−Pν−1)aνPνλν. so that forn ≥1
Tn−Tn−1 = pn PnPn−1
n
X
ν=1
Pν−1aνPνλν
= pn
PnPn−1 n−1
X
ν=1
∆(Pν−1Pνλν)sν +pnλnsn
=L1+L2, say.
Thus to prove the theorem it is sufficient to show that
∞
X
1
Pn pn
k−1
|Lν|k <∞, ν = 1,2.
Now
|∆(Pν−1Pνλν)| ≤pνPν|λν|+Pν|∆(Pνλν)|
≤CpνPν|λν| in view of (2.3). So
m+1
X
n=2
Pn pn
k−1
|L1|k=O(1)
m+1
X
n=2
pn PnPn−1k
n−1
X
ν=1
pνPν|λν||sν|
!k
=O(1)
m+1
X
n=2
pn
PnPn−1k
n−1
X
ν=1
(Pν|λν|)k|sν|kpν
! n−1 X
ν=1
pν
!k−1
=O(1)
m+1
X
n=2
pn
PnPn−1 n−1
X
ν=1
Pν|λν|pν|sν|k
=O(1)
m
X
ν=1
pν|λν|sν|k=O(1) in view of the lemma andPn|λn|=O(1).
Also
m+1
X
1
Pn pn
k−1
|L2|k =O(1)
m+1
X
1
pn|λn|k|sn|kPnk−1
=O(1)
m+1
X
1
pn|λn||sn|k
=O(1).
Thus a generalization of a corrected version of Theorem C is Theorem 2.2. Writing λn = µnPnthe conditions (1.5) and (1.4) become
(3.1)
∞
X
1
pn|µn|<∞,
(3.2) |∆(Pnµn)| ≤Cpn|µn|,
consequently Theorem D can be stated as:
IfP
an is bounded[ ¯N , Pn]k, k ≥ 1and{µn}is a sequence satisfying (3.1) and (3.2) then PanPnµnis summable|N , p¯ n|k, k ≥1.
Thus Theorem D is the same as Theorem 2.2 which is a generalization of the corrected version of Theorem C.
REFERENCES
[1] H. BOR, On|N , p¯ n|ksummability methods and|N , p¯ n|ksummability factors of infinite series, Ph.D.
Thesis (1982), Univ. of Ankara.
[2] H. BOR, On|N , p¯ n|ksummability factors, Proc. Amer. Math. Soc., 94 (1985), 419–422.
[3] H. BOR, On|N , p¯ n|ksummability factors of infinite series, Tamkang J. Math., 16 (1985), 13–20.
[4] M. ALI SARIGOL AND E. OZTURK, A note on|N , p¯ n|k summability factors of infinite series, Indian J. Math., 34 (1992), 167–171.
[5] N. SINGH, On|N , p¯ n|summability factors of infinite series, Indian J. Math., 10 (1968), 19–24.
[6] T. SINGH, A note on|N , p¯ n|summability factors of infinite series, J. Math. Sci., 12-13 (1977-78), 25–28.
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