Journal of Inequalities in Pure and
Applied Mathematics
A STUDY ON ALMOST INCREASING SEQUENCES
HÜSEYIN BOR
Department of Mathematics,
Erciyes University,
38039 Kayseri, Turkey.
EMail: bor@erciyes.edu.tr
URL: http://math.erciyes.edu.tr/hbor.htm
volume 4, issue 5, article 97,
2003.
Received 10 July, 2003;
accepted 18 August, 2003.
Communicated by: L. Leindler
Abstract
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Abstract
In this paper by using an almost increasing sequence a general theorem on
ϕ − | C, α |k summability factors, which generalizes some known results, has
been proved under weaker conditions.
2000 Mathematics Subject Classification: 40D15, 40F05, 40G05.
Key words: Absolute summability, Almost increasing sequences.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4
Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
References
A Study on Almost Increasing
Sequences
Hüseyin Bor
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1.
Introduction
P
Let (ϕn ) be a sequence of complex numbers and let
an be a given infinite
series with partial sums (sn ). We denote by σnα and tαn the n-th Cesáro means of
order α, with α > −1, of the sequences (sn ) and (nan ), respectively, i.e.,
σnα
(1.1)
n
1 X α−1
= α
A sv
An v=0 n−v
and
tαn
(1.2)
n
1 X α−1
A vav ,
= α
An v=1 n−v
A Study on Almost Increasing
Sequences
Hüseyin Bor
where
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Aαn = O(nα ), α > −1, Aα0 = 1 and Aα−n = 0 f or n > 0.
P
The series
an is said to be |C, α|k summable for k ≥ 1 and α > −1, if
(see [5])
Contents
(1.3)
∞
X
(1.4)
n
k−1
n=1
∞
X
α
1 αk
α k
σn − σn−1
=
|t | < ∞.
n n
n=1
and it is said to be |C, α; β| k summable for k ≥ 1, α > −1 and β ≥ 0, if (see
[6])
(1.5)
∞
X
n=1
n
βk+k−1
∞
X
α
α k
σn − σn−1
=
nβk−1 |tαn |k < ∞.
n=1
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The series
(see [2])
P
an is said to be ϕ−|C, α|k summable for k ≥ 1 and α > −1, if
∞
X
(1.6)
n−k |ϕn tαn |k < ∞.
n=1
1
1
In the special case when ϕn = n1− k (resp. ϕn = nβ+1− k ) ϕ−|C, α|k summability is the same as |C, α|k (resp. |C, α; β|k ) summability.
Bor [3] has proved the following theorem for ϕ−|C, 1|k summability factors
of infinite series.
Theorem 1.1. Let (Xn ) be a positive non-decreasing sequence and let (λn ) be
a sequence such that
|λn | Xn = O(1)
(1.7)
as
n→∞
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and
n
X
(1.8)
vXv ∆2 λv = O(1)
as
n → ∞.
v=1
If there exists an > 0 such that the sequence (n−k |ϕn |k ) is non-increasing
and
n
X
(1.9)
v −k |ϕv tv |k = O(Xn ) as
P
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n → ∞,
v=1
then the series
A Study on Almost Increasing
Sequences
an λn is ϕ−|C, 1|k summable for k ≥ 1.
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The aim of this paper is to generalize Theorem 1.1 under weaker conditions
for ϕ−|C, α|k summability. For this we need the concept of almost increasing
sequences. A positive sequence (bn ) is said to be almost increasing if there
exists a positive increasing sequence cn and two positive constants A and B
such that Acn ≤ bn ≤ Bcn (see [1]). Obviously every increasing sequence
is an almost increasing sequence but the converse need not be true as can be
n
seen from the example bn = ne(−1) . So we are weakening the hypotheses
of the theorem by replacing the increasing sequence with an almost increasing
sequence.
A Study on Almost Increasing
Sequences
Hüseyin Bor
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2.
Result
Now, we shall prove the following:
Theorem 2.1. Let (Xn ) be an almost increasing sequence and the sequence
(λn ) such that conditions (1.7) – (1.8) of Theorem 1.1 are satisfied. If there
exists an > 0 such that the sequence (n−k |ϕn |k ) is non-increasing and if the
sequence (wnα ), defined by (see [9])

α
α=1
 | |tn | ,
α
(2.1)
wn =
 max |tαv | , 0 < α < 1
1≤v≤n
satisfies the condition
(2.2)
m
X
n−k (wnα |ϕn |)k = O(Xm ) as
then the series
kα + > 1.
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m → ∞,
n=1
P
A Study on Almost Increasing
Sequences
an λn is ϕ−|C, α|k summable for k ≥ 1, 0 < α ≤ 1 and
We need the following lemmas for the proof of our theorem.
Lemma 2.2. ([4]). If 0 < α ≤ 1 and 1 ≤ v ≤ n, then
v
m
X
X
α−1 α−1
(2.3)
A
a
≤
max
A
a
p
p .
p=0 n−p 1≤m≤v p=0 m−p Contents
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Lemma 2.3. ([8]). If (Xn ) is an almost increasing sequence and the conditions
(1.7) and (1.8) of Theorem 1.1 are satisfied, then
(2.4)
m
X
Xn |∆λn | = O(1)
n=1
and
(2.5)
mXm |∆λm | = O(1),
m → ∞.
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Hüseyin Bor
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3.
Proof of Theorem 2.1
Let (Tnα ) be the n−th (C, α), with 0 < α ≤ 1, mean of the sequence (nan λn ).
Then, by (1.2), we have
Tnα
n
1 X α−1
= α
A vav λv .
An v=1 n−v
Using Abel’s transformation, we get
Tnα
n−1
v
n
X
1 X
λn X α−1
α−1
= α
∆λv
An−p pap + α
A vav ,
An v=1
An v=1 n−v
p=1
so that making use of Lemma 2.2, we have
n−1
v
n
X
|λ | X
X
1
n
α−1
|Tnα | ≤ α
|∆λv | Aα−1
pa
+
A
va
p
v
n−p
n−v
α
An An v=1
p=1
v=1
≤
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n−1
1 X α α
Av wv |∆λv | + |λn | wnα
α
An v=1
α
α
= Tn,1
+ Tn,2
,
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Since
α
α k α k
k α k
Tn,1 + Tn,2
≤2
Tn,1 + Tn,2 ,
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to complete the proof of the theorem, it is sufficient to show that
∞
X
n=1
α k
n−k ϕn Tn,r
< ∞ for
r = 1, 2,
by
(1.6).
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Now, when k > 1, applying Hölder’s inequality with indices k and k 0 , where
1
+ k10 = 1, we get that
k
m+1
X
α k
n−k ϕn Tn,1
n=2
≤
m+1
X
( n−1
)k
X
n−k (Aαn )−k |ϕn |k
Aαv wvα |∆λv |
n=2
= O(1)
m+1
X
v=1
( n−1
X
k
n−k n−αk |ϕn |
n=2
= O(1)
m
X
v αk (wvα )k |∆λv |
v=1
= O(1)
= O(1)
= O(1)
= O(1)
= O(1)
m
X
v=1
m
X
v=1
m
X
v=1
m
X
v=1
m−1
X
v=1
v αk (wvα )k |∆λv |
) ( n−1
)k−1
X
v αk (wvα )k |∆λv |
|∆λv |
v=1
m+1
X
v=1
n=v+1
m+1
X
n
n
−k
n=v+1
v
v
αk
αk
(wvα )k
(wvα )k
|∆λv | v
|∆λv | v
−k
−k
Hüseyin Bor
k
−k
|ϕn |
nαk
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k
|ϕn |
αk+
n
k
|ϕv |
k
m+1
X
|ϕv |
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v |∆λv | v −k (wvα |ϕv |)k
∆(v |∆λv |)
r=1
r
−k
(wrα
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dx
xαk+
v
v
X
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k
|ϕr | ) + O(1)m |∆λm |
m
X
v=1
v −k (wvα |ϕv |)k
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= O(1)
= O(1)
m−1
X
v=1
m−1
X
|∆(v |∆λv |)| Xv + O(1)mβm Xm
m−1
X
2 v ∆ λv Xv + O(1)
|∆λv+1 | Xv+1 + O(1)m |∆λm | Xm
v=1
= O(1)
as
v=1
m → ∞,
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.3.
Again, since |λn | = O(1/Xn ) = O(1), by (1.7), we have that
m
X
A Study on Almost Increasing
Sequences
Hüseyin Bor
α k
n−k ϕn Tn,2
n=1
=
m
X
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k−1
|λn |
|λn | n
−k
(wnα
k
|ϕn |)
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n=1
= O(1)
= O(1)
= O(1)
m
X
n=1
m−1
X
n=1
m−1
X
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|λn | n−k (wnα |ϕn |)k
∆ |λn |
n
X
v
−k
(wvα
k
|ϕv |) + O(1) |λm |
v=1
|∆λn | Xn + O(1) |λm | Xm = O(1)
m
X
n−k (wnα |ϕn |)k
n=1
as
n=1
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.3.
m → ∞,
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Therefore, we get that
m
X
α k
n−k ϕn Tn,r
= O(1)
as
m → ∞,
for r = 1, 2.
n=1
This completes the proof of the theorem.
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Hüseyin Bor
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4.
Special Cases
1. If we take (Xn ) as a positive non-decreasing sequence, α = 1 and ϕn =
1
n1− k in Theorem 2.1, then we get Theorem 1.1.
2. If we take (Xn ) as a positive non-decreasing sequence, α = 1 and ϕn =
1
n1− k in Theorem 2.1, then we get a result due to Mazhar [7] for |C, 1|k
summability factors of infinite series.
1
1
3. If we take = 1 and ϕn = n1− k (resp. = 1 and ϕn = nβ+1− k ), then we
get a new result related to |C, α|k (resp. |C, α; β|k ) summability factors.
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References
[1] S. ALJANCIC AND D. ARANDELOVIC, O-regularly varying functions,
Publ. Inst. Math., 22 (1977), 5–22.
[2] M. BALCI, Absolute ϕ-summability factors, Comm. Fac. Sci. Univ. Ankara,
Ser. A1 , 29 (1980), 63–80.
[3] H. BOR, Absolute summability factors, Atti Sem. Mat. Fis. Univ. Modena,
39 (1991), 419–422.
[4] L.S. BOSANQUET, A mean value theorem, J. London Math. Soc., 16
(1941), 146–148.
[5] T.M. FLETT, On an extension of absolute summability and some theorems
of Littlewood and Paley, Proc. London Math. Soc., 7 (1957), 113–141.
[6] T.M. FLETT, Some more theorems concerning the absolute summability of
Fourier series, Proc. London Math. Soc., 8 (1958), 357–387.
[7] S.M. MAZHAR, On |C, 1|k summability factors, Indian J. Math., 14
(1972), 45–48.
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[8] S.M. MAZHAR, Absolute summability factors of infinite series, Kyungpook Math. J., 39 (1999), 67–73.
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[9] T. PATI, The summability factors of infinite series, Duke Math. J., 21
(1954), 271–284.
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