Journal of Inequalities in Pure and
Applied Mathematics
A NOTE ON ABSOLUTE NÖRLUND SUMMABILITY
HÜSEYİN BOR
Department of Mathematics
Erciyes University
38039 Kayseri, Turkey
volume 6, issue 4, article 115,
2005.
Received 26 September, 2005;
accepted 17 October, 2005.
Communicated by: L. Leindler
EMail: bor@erciyes.edu.tr
Abstract
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c 2000 Victoria University
ISSN (electronic): 1443-5756
287-05
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Abstract
In this paper a main theorem on |N, pn |k summability factors, which generalizes
a result of Bor [2] on |N, pn | summability factors, has been proved.
2000 Mathematics Subject Classification: 40D15, 40F05, 40G05.
Key words: Nörlund summability, summability factors, power increasing sequences.
A Note on Absolute Nörlund
Summability
The author is grateful to the referee for his valuable suggestions for the improvement
of this paper.
Hüseyin Bor
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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3
6
7
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1.
Introduction
A positive sequence (bn ) is said to be almost increasing if there exist a positive
increasing sequence (cn ) and two positive constants A and B such that Acn ≤
bn ≤ Bcn (see [1]). A positive sequence (γn ) is said to be a quasi β-power
increasing sequence if there exists a constant K = K(β, γ) ≥ 1 such that
Knβ γn ≥ mβ γm
(1.1)
holds for all n ≥ m ≥ 1. 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 the example, say γn = n−β
for β > 0. We denote by BVO the BV ∩ CO , where CO and BV are the null
sequences
P and sequences with bounded variation, respectively.
Let
an be a given infinite series with the sequence of partial sums (sn )
and wn = nan . By uαn and tαn we denote the n-th Cesàro means of order α, with
α > −1, of thePsequences (sn ) and (wn ), respectively.
The series an is said to be summable |C, α|k , k ≥ 1, if (see [4])
(1.2)
∞
X
nk−1 uαn − uαn−1
k
=
n=1
∞
X
1 αk
|tn | < ∞.
n
n=1
Let (pn ) be a sequence of constants, real or complex, and let us write
(1.3)
Pn = p0 + p1 + p2 + · · · + pn 6= 0, (n ≥ 0).
The sequence-to-sequence transformation
(1.4)
n
1 X
σn =
pn−v sv
Pn v=0
A Note on Absolute Nörlund
Summability
Hüseyin Bor
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defines the sequence (σn ) of the Nörlund mean ofP
the sequence (sn ), generated
by the sequence of coefficients (pn ). The series
an is said to be summable
|N, pn |k , k ≥ 1, if (see [3])
(1.5)
∞
X
nk−1 |σn − σn−1 |k < ∞.
n=1
In the special case when
(1.6)
pn =
Γ(n + α)
, α≥0
Γ(α)Γ(n + 1)
the Nörlund mean reduces to the (C, α) mean and |N, pn |k summability becomes |C, α|k summability. For pn = 1 and Pn = n, we get the (C, 1) mean
and then |N, pn |k summability becomes |C, 1|k summability. For any sequence
(λn ), we write ∆λn = λn − λn+1 .
The known results. Concerning the |C, 1|k and |N, pn |k summabilities Varma
[6] has proved the following theorem.
Theorem
A. Let p0 > 0, pn ≥ 0 and (pn )P
be a non-increasing sequence.
P
If
an is summable |C, 1|k , then the series
an Pn (n + 1)−1 is summable
|N, pn |k , k ≥ 1.
Quite recently Bor [2] has proved the following theorem.
Theorem B. Let (pn ) be as in Theorem A, and let (Xn ) be a quasi β-power
increasing sequence with some 0 < β < 1. If
n
X
1
(1.7)
|tv | = O(Xn ) as n → ∞,
v
v=1
A Note on Absolute Nörlund
Summability
Hüseyin Bor
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and the sequences (λn ) and (βn ) satisfy the following conditions
(1.8)
Xn λn = O(1),
(1.9)
|∆λn | ≤ βn ,
βn → 0,
(1.10)
A Note on Absolute Nörlund
Summability
Hüseyin Bor
X
(1.11)
nXn |∆βn | < ∞,
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then the series
P
an Pn λn (n + 1)
−1
is summable |N, pn |.
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2.
Main Result
The aim of this paper is to generalize Theorem B for |N, pn |k summability. Now
we shall prove the following theorem.
Theorem 2.1. Let (pn ) be as in Theorem A, and let (Xn ) be a quasi β-power
increasing sequence with some 0 < β < 1. If
n
X
1
(2.1)
|tv |k = O(Xn ) as n → ∞,
v
v=1
and the sequences (λn ) and (βn ) satisfy the conditions from (1.8) to (1.11) of
Theorem B; further suppose that
P
an Pn λn (n + 1)−1 is summable |N, pn |k , k ≥ 1.
Remark 1. It should be noted that if we take k = 1, then we get Theorem B. In
this case condition (2.2) is not needed.
We need the following lemma for the proof of our theorem.
Lemma 2.2 ([5]). Except for the condition (2.2), under the conditions on (Xn ),
(λn ) and (βn ) as taken in the statement of the theorem, the following conditions
hold when (1.11) is satisfied:
(2.3)
(2.4)
Hüseyin Bor
(λn ) ∈ BVO ,
(2.2)
then the series
A Note on Absolute Nörlund
Summability
nβn Xn = O(1) as n → ∞,
∞
X
n=1
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βn Xn < ∞.
J. Ineq. Pure and Appl. Math. 6(4) Art. 115, 2005
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3.
Proof of Theorem 2.1
In order to prove the theorem, we need consider
P only the special case in which
(N, pn ) is (C, 1), that is, we shall prove that an λn is summable |C, 1|k . Our
theorem will then follow by means of Theorem A. Let Tn be the n−th (C, 1)
mean of the sequence (nan λn ), that is,
n
1 X
Tn =
vav λv .
n + 1 v=1
(3.1)
A Note on Absolute Nörlund
Summability
Using Abel’s transformation, we have
n
n−1
1 X
1 X
vav λv =
∆λv (v + 1)tv + λn tn
Tn =
n + 1 v=1
n + 1 v=1
= Tn,1 + Tn,2 , say.
∞
X
1
|Tn,r |k < ∞
n
n=1
for r = 1, 2, by (1.2).
n=2
1
|Tn,1 |k ≤
n
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Now, we have that
m+1
X
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To complete the proof of the theorem, it is sufficient to show that
(3.2)
Hüseyin Bor
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m+1
X
n=2
1
n(n + 1)k
= O(1)
m+1
X
n=2
( n−1
Xv+1
v=1
( n−1
X
1
nk+1
v=1
v
)k
v |∆λv | |tv |
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)k
v |∆λv | |tv |
J. Ineq. Pure and Appl. Math. 6(4) Art. 115, 2005
http://jipam.vu.edu.au
= O(1)
= O(1)
= O(1)
m+1
X
n=2
m+1
X
n=2
m+1
X
n=2
m
X
1
n2
1
n2
1
n2
( n−1
X
)
v |∆λv | |tv |k
v=1
n−1
X
v |∆λv | |tv |k
v=1
( n−1
X
(
×
)k−1
n−1
1X
v |∆λv |
n v=1
(by (2.2))
)
vβv |tv |k
(by (1.9))
v=1
m
X
|tv |k
1
= O(1)
vβv
= O(1)
vβv |tv |
n2
v
v=1
v=1
n=v+1
= O(1)
= O(1)
= O(1)
= O(1)
m−1
X
v=1
m−1
X
v=1
m−1
X
v=1
m−1
X
k
∆(vβv )
m+1
X
v
X
|tr |k
r=1
+ O(1)mβm
Hüseyin Bor
m
X
|tv |k
v=1
v
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|∆(vβv )| Xv + O(1)mβm Xm
(by (2.1))
|(v + 1)∆βv − βv | Xv + O(1)mβm Xm
v |∆βv | Xv + O(1)
v=1
= O(1)
r
A Note on Absolute Nörlund
Summability
as m → ∞,
in view of (1.11), (2.3) and (2.4).
m−1
X
v=1
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|βv | Xv + O(1)mβm Xm
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J. Ineq. Pure and Appl. Math. 6(4) Art. 115, 2005
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Again
m
m
k
X
X
1
k
k |tn |
|Tn,2 | =
|λn |
n
n
n=1
n=1
=
m
X
|λn |k−1 |λn |
n=1
= O(1)
= O(1)
= O(1)
m−1
X
n=1
m−1
X
n=1
m−1
X
∆ |λn |
m
X
|tn |k
|tn |k
= O(1)
|λn |
n
n
n=1
n
X
|tv |k
v=1
v
+ O(1) |λm |
|∆λn | Xn + O(1) |λm | Xm
(by (2.2))
m
X
|tn |k
n=1
A Note on Absolute Nörlund
Summability
n
Hüseyin Bor
(by (2.1))
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βn Xn + O(1) |λm | Xm = O(1)
as m → ∞,
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n=1
by virtue of (1.8) and (2.4). This completes the proof of the theorem.
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References
[1] S. ALJANČIĆ AND D. ARANDELOVIĆ, O-regularly varying functions,
Publ. Inst. Math., 22 (1977), 5–22.
[2] H. BOR, Absolute Nörlund summability factors, J. Inequal. Pure Appl.
Math., 6(3) (2005), Art. 62. [ONLINE http://jipam.vu.edu.au/
article.php?sid=535].
[3] D. BORWEIN AND F.P. CASS, Strong Nörlund summability, Math. Zeith.,
103 (1968), 94–111.
[4] T.M. FLETT, On an extension of absolute summability and some theorems
of Littlewood and Paley, Proc. London Math. Soc., 7 (1957), 113–141.
[5] L. LEINDLER, A new application of quasi power increasing sequences,
Publ. Math. Debrecen, 58 (2001), 791–796.
[6] R.S. VARMA, On the absolute Nörlund summability factors, Riv. Math.
Univ. Parma (4), 3 (1977), 27–33.
A Note on Absolute Nörlund
Summability
Hüseyin Bor
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