Journal of Inequalities in Pure and
Applied Mathematics
AN APPLICATION OF ALMOST INCREASING AND
DELTA-QUASI-MONOTONE SEQUENCES
volume 1, issue 2, article 18,
2000.
H. BOR
Department of Mathematics
Erciyes University
Kayseri 38039, TURKEY
EMail: bor@erciyes.edu.tr
Received 14 April, 2000;
accepted 27 April, 2000.
Communicated by: L. Leindler
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
In this paper a general theorem on absolute weighted mean summability factors
has been proved under weaker conditions by using an almost increasing and
δ-quasi-monotone sequences.
2000 Mathematics Subject Classification: 40D15, 40F05
Key words: Almost Increasing Sequences, Quasi-monotone Sequences, Absolute
Summability Factors, Infinite Series
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Main Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
H. Bor
3
6
8
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1.
Introduction
A sequence (bn ) of positive numbers is said to be δ-quasi-monotone, if bn > 0
ultimatelyP
and ∆bn ≥ −δn , where (δn ) is a sequence of positive numbers (see
an be a given infinite series with (sn ) as the sequence of its n-th
[2]). Let
partial sums. By un and tn we denote the
Pn-th (C, 1) means of the sequence
(sn ) and (nan ), respectively. The series
an is said to be summable |C, 1|k ,
k ≥ 1, if (see [5])
∞
X
n=1
n
k−1
∞
X
1
|un − un−1 | =
|tn |k < ∞.
n
n=1
k
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
H. Bor
Let (pn ) be a sequence of positive numbers such that
Pn =
n
X
pv → ∞ as n → ∞, (P−i = p−i = 0, i ≥ 1) .
v=0
The sequence-to-sequence transformation
n
1 X
p v sv
zn =
Pn v=0
defines the sequence (zn ) of the N̄ , pn mean of the sequence
P (sn ) generated
by the sequence
of
an is said to be
coefficients (pn ) (see [6]). The series
summable N̄ , pn k , k ≥ 1, if (see [3])
k−1
∞ X
Pn
|zn − zn−1 |k < ∞.
pn
n=1
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In the special case pn = 1 for all values of n (resp. k = 1), then N̄ , pn k
Also if we take
summability is the
same
as |C, 1|k (resp. N̄ , pn ) summability.
1
1 pn = n+1 , then N̄ , pn k summability reduces to N̄ , n+1 k summability.
Mazhar [7] has proved the following theorem for summability factors by
using δ-quasi-monotone sequences.
Theorem 1.1. Let λn → 0 as n → ∞. Suppose that there
P exists a sequence
of
numbers
(A
)
such
that
it
is
δ-quasi-monotone
with
nδn log n < ∞,
n
P
An log n is convergent and |∆λn | ≤ |An | for all n. If
m
X
1
|tn |k = O (log m) as m → ∞,
n
n=1
H. Bor
P
an λn is summable |C, 1|k , k ≥ 1.
Later on Bor [4] generalized Theorem 1.1 for a N̄ , pn k summability method
in the following form.
then the series
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
Theorem 1.2. Let λn → 0 as n → ∞ and let (pn ) be a sequence of positive
numbers such that
Pn = O (npn ) as n → ∞.
Suppose that there
P exists a sequence
P of numbers (An ) such that it is δ-quasimonotone with
nδn Xn < ∞,
An Xn is convergent and |∆λn | ≤ |An | for
all n. If
m
X
pn
|tn |k = O (Xm ) as m → ∞,
P
n=1 n
P
where
(X
)
is
a
positive
increasing
sequence,
then
the
series
an λn is summable
n
N̄ , pn , k ≥ 1.
k
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It should be noted that if we take Xn = log n and pn = 1 for all values of n
in Theorem 1.2, then we get Theorem 1.1.
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
H. Bor
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2.
The Main Result.
1
Due to the restriction Pn = O (npn ) on (pn ) , no result for pn = n+1
can be deduced from Theorem 1.2. Therefore the aim of this paper is to prove Theorem
1.2 under weaker conditions and in a more general form without this condition. For this we need the concept of almost increasing sequence. A positive
sequence (dn ) is said to be almost increasing if there exists a positive increasing
sequence (cn ) and two positive constants A and B such that Acn ≤ dn ≤ Bcn
(see [1]). Obviously, every increasing sequence is almost increasing but the
n
converse need not be true as can be seen from the example dn = ne(−1) . Since
(Xn ) is increasing in Theorem 1.2, we are weakening the hypotheses of the theorem by replacing the increasing sequence with an almost increasing sequence.
Now, we shall prove the following theorem.
Theorem 2.1. Let (Xn ) be an almost increasing sequence and λn → 0 as n →
∞. Suppose thatPthere exists a sequence
of numbers (An ) such that it is δ-quasiP
monotone with
nδn Xn < ∞,
An Xn is convergent and |∆λn | ≤ |An | for
all n. If
m
X
1
|λn | = O (1) ,
n
n=1
m
X
1
|tn |k = O (Xm ) as m → ∞
n
n=1
and
m
X
pn
|tn |k = O (Xm ) as m → ∞,
P
n=1 n
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
H. Bor
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then the series
P
an λn is summable N̄ , pn k , k ≥ 1.
We need the following lemmas for the proof of our theorem.
Lemma 2.2. Under the conditions of the theorem, we have
|λn | Xn = O (1) as n → ∞.
Proof. Since λn → 0 as n → ∞, we have that
∞
∞
∞
∞
X
X
X
X
|λn | Xn = Xn ∆λv ≤ Xn
|∆λv | ≤
Xv |∆λv | ≤
Xv |Av | < ∞.
v=n
v=n
v=0
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
v=0
H. Bor
Hence |λn | Xn = O (1) as n → ∞.
Lemma 2.3. Let
P(Xn ) be an almost
P increasing sequence. If (An ) is δ-quasimonotone with nδn Xn < ∞, An Xn is convergent, then
nAn Xn = O (1) ,
∞
X
nXn |∆An | < ∞.
n=1
The proof of Lemma 2.3 is similar to the proof of Theorem 1 and Theorem
2 of Boas [2, case γ = 1], and hence is omitted.
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3.
Proof of the Theorem
P
Proof of Theorem 2.1. Let (Tn ) denote the N̄ , pn mean of the series an λn .
Then, by definition and changing the order of summation, we have
n
v
n
1 X X
1 X
ai λ i =
Tn =
pv
(Pn − Pv−1 ) av λv .
Pn v=0 i=0
Pn v=0
Then, for n ≥ 1, we have
Tn − Tn−1
n
n
pn X Pv−1 λv
pn X
=
Pv−1 av λv =
vav .
Pn Pn−1 v=1
Pn Pn−1 v=1
v
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
H. Bor
By Abel’s transformation, we get
Tn − Tn−1
n−1
n+1
pn X
v+1
=
pn tn λn −
pv tv λv
nPn
Pn Pn−1 v=1
v
+
pn
Pn Pn−1
n−1
X
v=1
Pv tv ∆λv
v+1
pn
+
v
Pn Pn−1
= Tn,1 + Tn,2 + Tn,3 + Tn,4 ,
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n−1
X
v=1
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Pv tv λv+1
1
v
say.
Since
|Tn,1 + Tn,2 + Tn,3 + Tn,4 |k ≤ 4k |Tn,1 |k + |Tn,2 |k + |Tn,3 |k + |Tn,4 |k ,
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to complete the proof of the theorem, it is enough to show that
k−1
∞ X
Pn
|Tn,r |k < ∞ for r = 1, 2, 3, 4.
pn
n=1
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Since λn is bounded by the hypothesis, we have that
k−1
m X
Pn
n=1
pn
k
|Tn,1 |
m
X
pn
= O (1)
|tn |k |λn | |λn |k−1
P
n=1 n
= O (1)
m
X
pn
|tn |k |λn |
P
n=1 n
m
n
X
X
pn
pv
k
= O (1)
|∆λn |
|tv | + O (1) |λm |
|tn |k
P
P
n=1
v=1 v
n=1 n
m−1
X
= O (1)
= O (1)
m−1
X
n=1
m−1
X
|∆λn | Xn + O (1) |λm | Xm
H. Bor
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|An | Xn + O (1) |λm | Xm
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n=1
= O (1) as m → ∞,
by virtue of the hypotheses of the theorem and Lemma 2.2.
Now, when k > 1, applying Hölder’s inequality, as in Tn,1 , we have that
m+1
X
n=2
Pn
pn
k−1
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|Tn,2 |k
= O (1)
An Application of Almost
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δ-Quasi-Monotone Sequences
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m+1
X
n=2
n−1
pn X
pv |tv |k |λv |k ×
Pn Pn−1 v=1
(
1
n−1
X
Pn−1
v=1
)k−1
Page 9 of 14
pv
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= O (1)
= O (1)
m
X
v=1
m
X
k
k−1
pv |tv | |λv | |λv |
|λv |
v=1
m+1
X
pn
P P
n=v+1 n n−1
pv
|tv |k = O (1) as m → ∞.
Pv
Again we have that,
m+1
X
n=2
Pn
pn
k−1
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
|Tn,3 |k
= O (1)
m+1
X
n=2
m
X
pn
Pn Pn−1
n−1
X
(
Pv |tv |k |Av | ×
v=1
1
n−1
X
Pn−1
v=1
)k−1
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m+1
X
pn
= O (1)
Pv |tv | |Av |
P P
v=1
n=v+1 n n−1
= O (1)
m
X
k
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|tv |k |Av |
v=1
= O (1)
= O (1)
m
X
v=1
m−1
X
v=1
v |Av |
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i=1
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|tv |k
v
∆ (v |Av |)
H. Bor
Pv |Av |
i
k
|ti | + O (1) m |Am |
m
X
1
v=1
v
|tv |k
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= O (1)
= O (1)
m−1
X
v=1
m−1
X
|∆ (v |Av |)| Xv + O (1) m |Am | Xm
v |∆Av | Xv + O (1)
v=1
m−1
X
|Av+1 | Xv+1 + O (1) m |Am | Xm
v=1
= O (1) as m → ∞,
in view of the hypotheses of Theorem 2.1 and Lemma 2.3.
Finally, we get that
m+1
X
n=2
Pn
pn
k−1
≤
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
|Tn,4 |k
m+1
X
n=2
H. Bor
n−1
pn X
1
Pv |tv |k |λv+1 | ×
Pn Pn−1 v=1
v
m
X
(
1
Pn−1
n−1
X
1
Pv |λv+1 |
v
v=1
= O (1)
m
X
v=1
m−1
X
v=1
k
|λv+1 |
1
|tv |k
v
∆ |λv+1 |
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1
i=1
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m+1
1 X
pn
= O (1)
Pv |tv | |λv+1 |
v n=v+1 Pn Pn−1
v=1
= O (1)
)k−1
i
|ti |k + O (1) |λm+1 |
m
X
1
v=1
v
|tv |k
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= O (1)
= O (1)
m−1
X
v=1
m−1
X
|∆λv+1 | Xv+1 + O (1) |λm+1 | Xm+1
|λv+1 | Xv+1 + O (1) |λm+1 | Xm+1
v=1
= O (1) as m → ∞,
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2.
Therefore we get
k−1
m X
Pn
n=1
pn
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
|Tn,r |k = O (1) as m → ∞, for r = 1, 2, 3, 4.
This completes the proof of the theorem.
1
If we take pn = 1 for all values
of n (resp. pn = n+1
), then we get a result
1 concerning the |C, 1|k (resp. N̄ , n+1 k ) summability factors.
H. Bor
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References
[1] L.S. ALJANCIC AND D. ARANDELOVIC, O- regularly varying functions,
Publ. Inst. Math., 22 (1977), 5–22.
[2] R. P. BOAS, Quasi positive sequences and trigonometric series, Proc. London Math. Soc., 14A (1965), 38–46.
[3] H. BOR, On two summability methods, Math. Proc.Camb. Philos. Soc., 97
(1985), 147–149.
[4] H. BOR, On quasi monotone sequences and their applications, Bull. Austral.
Math. Soc., 43 (1991), 187–192.
An Application of Almost
Increasing and
δ-Quasi-Monotone Sequences
H. Bor
[5] T. M. FLETT, On an extension of absolute summability and some theorems
of Littlwood and Paley, Proc. London Math. Soc., 7 (1957), 113–141.
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[6] G. H. HARDY, Divergent Series, Oxford Univ. Press, 1949.
Contents
[7] S. M. MAZHAR, On generalized quasi-convex sequence and its applications, Indian J. Pure Appl. Math., 8 (1977), 784–790.
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