Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 4, Article 66, 2003
INCLUSION THEOREMS FOR ABSOLUTE SUMMABILITY METHODS
H. S. ÖZARSLAN
D EPARTMENT OF M ATHEMATICS ,
E RCIYES U NIVERSITY,
38039 K AYSERI ,
T URKEY.
seyhan@erciyes.edu.tr
Received 07 April, 2003; accepted 30 April, 2003
Communicated by L. Leindler
A BSTRACT. In this paper we have proved two theorems concerning an inclusion between two
absolute summability methods by using any absolute summability factor.
Key words and phrases: Absolute summability, Summability factors, Infinite series.
2000 Mathematics Subject Classification. 40D15, 40F05, 40G99.
1. I NTRODUCTION
P
Let
an be a given infinite series with partial sums (sn ), and rn = nan . By un and
Ptn we
denote the n-th (C, 1) means of the sequences (sn ) and (rn ), respectively. The series
an is
said to be summable |C, 1|k , k ≥ 1, if (see [4])
∞
X
(1.1)
nk−1 |un − un−1 |k < ∞.
n=1
But since tn = n(un − un−1 ) (see [7]), the condition (1.1) can also be written as
∞
X
1
|tn |k < ∞.
n
n=1
(1.2)
The series
P
an is said to be summable |C, 1; δ|k k ≥ 1 and δ ≥ 0, if (see [5])
(1.3)
∞
X
nδk−1 |tn |k < ∞.
n=1
If we take δ = 0, then |C, 1; δ|k summability is the same as |C, 1|k summability.
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
046-03
2
H. S. Ö ZARSLAN
Let (pn ) be a sequence of positive numbers such that
Pn =
(1.4)
n
X
pv → ∞ as n → ∞,
(P−i = p−i = 0, i ≥ 1).
v=0
The sequence-to-sequence transformation
Tn =
(1.5)
n
1 X
p v sv
Pn v=0
defines the sequence (Tn ) of the (N̄ , pn ) mean of the sequence (sn ), generated by the sequence
of coefficientsP(pn ) (see [6]).
The series an is said to be summable N̄ , pn k , k ≥ 1, if (see [1])
k−1
∞ X
Pn
(1.6)
|∆Tn−1 |k < ∞
pn
n=1
and it is said to be summable N̄ , pn ; δ k , k ≥ 1 and δ ≥ 0, if (see [3])
δk+k−1
∞ X
Pn
(1.7)
|∆Tn−1 |k < ∞,
pn
n=1
where
∆Tn−1 = −
(1.8)
n
pn X
Pv−1 av ,
Pn Pn−1 v=1
n ≥ 1.
N̄ , pn ; δ summability is the
In the special
case
when
δ
=
0
(resp.
p
=
1
for
all
values
of
n)
n
k
same as N̄ , pn k (resp. |C, 1; δ|k ) summability.
Concerning inclusion relations between |C, 1|k and N̄ , pn k summabilities, the following
theorems are known.
Theorem 1.1. ([1]).Let k ≥ 1 and let (pn ) be a sequence of positive numbers such that as
n→∞
(i) Pn = O(npn ),
(1.9)
If the series
P
(ii) npn = O(Pn ).
an is summable |C, 1|k , then it is also summable N̄ , pn k .
Theorem 1.2. ([2]). Let k ≥ 1 and let (pn ) be a sequence
of positive numbers
such that
P
condition (1.9) of Theorem 1.1 is satisfied. If the series
an is summable N̄ , pn k , then it is
also summable |C, 1|k .
2. T HE M AIN R ESULT
The aim of this paper is to generalize the above theorems for |C, 1; δ|k and N̄ , pn ; δ k summabilities, by using a summability factors. Now, we shall prove the following theorems.
Theorem 2.1. Let k ≥ 1 and 0 ≤ δk < 1. Let (pn ) be a sequence of positive numbers such that
Pn = O(npn ) and
( )
δk−1
∞ δk
X
Pn
1
Pv
1
(2.1)
=O
.
pn
Pn−1
pv
Pv
n=v+1
J. Inequal. Pure and Appl. Math., 4(4) Art. 66, 2003
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I NCLUSION T HEOREMS FOR A BSOLUTE S UMMABILITY M ETHODS
3
P
P
Let
an be summable |C, 1; δ|k . Then
an λn is summable N̄ , pn ; δ k , if (λn ) satisfies the
following conditions:
1−δk
k
Pn
(2.2)
n∆λn = O
,
npn
λn = O
(2.3)
Pn
npn
k1
.
Theorem 2.2. Let k ≥ 1 and 0 ≤ δk < 1. Let (pn ) be P
a sequence of positive
numbers
such
. Then
that
np
=
O(P
)
and
satisfies
the
condition
(2.1).
Let
a
be
summable
N̄
,
p
;
δ
n
n
n
n
k
P
an λn is summable |C, 1; δ|k , if (λn ) satisfies the following conditions:
1−δk
k
npn
(2.4)
n∆λn = O
Pn
λn = O
(2.5)
npn
Pn
k1
.
Remark 2.3. It may be noted that, if we take λn = 1 and δ = 0 in Theorem 2.1 and Theorem
2.2, then we get Theorem 1.1 and Theorem 1.2, respectively. In this case condition (2.1) reduces
to
m+1
m+1
X
X 1
pn
1
1
=
−
=O
as m → ∞,
P P
Pn−1 Pn
Pv
n=v+1 n n−1
n=v+1
which always holds.
Proof of Theorem 2.1. Since
n
1 X
tn =
vav
n + 1 v=1
we have that
n+1
tn − tn−1 .
n P
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
Tn =
pv
ai λ i =
(Pn − Pv−1 )av λv .
Pn v=0 i=0
Pn v=0
an =
Then, for n ≥ 1, we have
Tn − Tn−1
n
pn X
Pv−1 av λv .
=
Pn Pn−1 v=1
By Abel’s transformation, we have
Tn − Tn−1 =
n−1
n−1
pn X
v+1
pn X
v+1
Pv ∆λv
tv −
p v λv
tv
Pn Pn−1 v=1
v
Pn Pn−1 v=1
v
n−1
p n X Pv
pn n + 1
+
λv+1 tv +
λn
tn
Pn Pn−1 v=1 v
Pn
n
= Tn,1 + Tn,2 + Tn,3 + Tn,4 ,
J. Inequal. Pure and Appl. Math., 4(4) Art. 66, 2003
say.
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4
H. S. Ö ZARSLAN
Since
|Tn,1 + Tn,2 + Tn,3 + Tn,4 |k ≤ 4k (|Tn,1 |k + |Tn,2 |k + |Tn,3 |k + |Tn,4 |k ),
to complete the proof of the theorem, it is enough to show that
δk+k−1
∞ X
Pn
(2.6)
pn
n=1
|Tn,r |k < ∞
for
r = 1, 2, 3, 4.
Now, when k > 1, applying Hölder’s inequality with indices k and k 0 , where
get
m+1
X
n=2
≤
Pn
pn
δk+k−1
m+1
X
n=2
= O(1)
Pn
pn
n=2
= O(1)
m+1
X
n=2
= O(1)
m+1
X
n=2
= O(1)
= O(1)
m
X
v=1
m
X
= O(1)
1
k0
= 1, we
Pn
pn
!k
v+1
Pv
|tv | |∆λv |
v
v=1
n−1
X
1
k
Pn−1
δk−1
n−1
X
Pv
|tv | |v∆λv |
pv
vpv
v=1
!k
n−1
X
|tv | |v∆λv | pv
1
k
Pn−1
Pn
pn
δk−1
Pn
pn
δk−1
1
k
Pn−1
Pn−1
|tv |k |v∆λv |k pv
|tv |k |v∆λv |k pv
v=1
m+1
X
n=v+1
k
k
|tv | |v∆λv |
!k
v=1
n−1
X
1
v=1
m
X
+
|Tn,1 |k
δk−1
m+1
X
1
k
Pv
pv
Pn
pn
δk−1
1
n−1
X
Pn−1
v=1
1
n−1
X
Pn−1
v=1
!k−1
pv
1
Pn−1
δk−1
v δk−1 |tv |k = O(1)
m → ∞.
as
v=1
by virtue of the hypotheses of Theorem 2.1.
Again using Hölder’s inequality,
m+1
X
n=2
Pn
pn
δk+k−1
= O(1)
m+1
X
n=2
= O(1)
|Tn,2 |k
m
X
Pn
pn
1
n−1
X
Pn−1
v=1
δk−1
k
k
pv |tv | |λv |
v=1
J. Inequal. Pure and Appl. Math., 4(4) Art. 66, 2003
m+1
X
n=v+1
pv |tv |k |λv |k
Pn
pn
δk−1
!k−1
pv
1
Pn−1
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I NCLUSION T HEOREMS FOR A BSOLUTE S UMMABILITY M ETHODS
= O(1)
= O(1)
δk−1
m X
Pv
v=1
m
X
pv
5
|tv |k |λv |k
v δk−1 |tv |k |λv |k
v=1
= O(1)
m
X
v δk−1 |tv |k = O(1)
as m → ∞,
v=1
by virtue of the hypotheses of Theorem 2.1.
Also
m+1
X Pn δk+k−1
|Tn,3 |k
pn
n=2
≤
m+1
X
n=2
= O(1)
Pn
pn
δk−1
m+1
X
n=2
= O(1)
m+1
X
n=2
= O(1)
= O(1)
= O(1)
m
X
v=1
m
X
v=1
m
X
Pn
pn
Pn
pn
!k
|λv+1 |
|tv |
Pv
v
v=1
n−1
X
1
k
Pn−1
δk−1
!k
|λv+1 |
|tv |
vpv
v
v=1
n−1
X
1
k
Pn−1
1
n−1
X
Pn−1
v=1
δk−1
k
m+1
X
k
pv |λv+1 | |tv |
|tv |
Pv
pv
δk−1
Pn
pn
δk−1
1
n
X
Pn−1
v=1
k
pv |λv+1 | |tv |
n=v+1
k
k
!k−1
pv
1
Pn−1
|λv+1 |k
v δk−1 |tv |k = O(1)
as
m → ∞,
v=1
by virtue of the hypotheses of Theorem 2.1.
Lastly
δk+k−1
δk−1
m m X
X
Pn
Pn
k
|Tn,4 | = O(1)
|λn |k |tn |k
p
p
n
n
n=1
n=1
m
X Pn δk−1
= O(1)
|λn |k |tn |k nδk−1
np
n
n=1
= O(1)
m
X
nδk−1 |tn |k = O(1)
as
m → ∞,
n=1
by virtue of the hypotheses of Theorem 2.1.
This completes the proof of Theorem 2.1.
Proof of Theorem 2.2. Let (Tn ) denotes the (N̄ , pn ) mean of the series
P
an . We have
n
n
1 X
1 X
Tn =
p v sv =
(Pn − Pv−1 )av .
Pn v=0
Pn v=0
J. Inequal. Pure and Appl. Math., 4(4) Art. 66, 2003
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6
H. S. Ö ZARSLAN
Since
Tn − Tn−1 =
n
pn X
Pv−1 av ,
Pn Pn−1 v=0
we have that
an =
(2.7)
−Pn
Pn−2
∆Tn−1 +
∆Tn−2 .
pn
pn−1
Let
n
1 X
tn =
vav λv .
n + 1 v=1
By using (2.7) we get
n
1 X
−Pv
Pv−2
tn =
v
∆Tv−1 +
∆Tv−2 λv
n + 1 v=1
pv
pv−1
n−1
n
1 X
Pv
nPn λn
1 X Pv−2
=
(−v) ∆Tv−1 λv −
∆Tn−1 +
v
∆Tv−2 λv
n + 1 v=1
pv
(n + 1)pn
n + 1 v=1 pv−1
n−1
n−1
1 X
Pv
1 X
Pv−1
nPn λn
(−v) ∆Tv−1 λv +
(v + 1)
∆Tv−1 λv+1 −
∆Tn−1
=
n + 1 v=1
pv
n + 1 v=1
pv
(n + 1)pn
n−1
nPn λn
1 X ∆Tv−1
{−vλv Pv + (v + 1)λv+1 Pv−1 } −
∆Tn−1
=
n + 1 v=1 pv
(n + 1)pn
n−1
=
nPn λn
1 X ∆Tv−1
{−vλv Pv + (v + 1)λv+1 (Pv − pv )} −
∆Tn−1
n + 1 v=1 pv
(n + 1)pn
n−1
nPn λn
1 X ∆Tv−1
{−vλv Pv + (v + 1)λv+1 Pv − (v + 1)λv+1 pv } −
∆Tn−1
=
n + 1 v=1 pv
(n + 1)pn
n−1
1 X ∆Tv−1
nPn λn
=
{−(∆vλv )Pv − (v + 1)λv+1 pv } −
∆Tn−1
n + 1 v=1 pv
(n + 1)pn
n−1
n−1
1 X −Pv
1 X
=
∆Tv−1 {v∆λv − λv+1 } −
∆Tv−1 (v + 1)λv+1
n + 1 v=1 pv
n + 1 v=1
nPn λn
∆Tn−1
(n + 1)pn
n−1
n−1
1 X vPv
1 X Pv
=−
∆λv ∆Tv−1 +
∆Tv−1 λv+1
n + 1 v=1 pv
n + 1 v=1 pv
−
n−1
1 X
nPn λn
−
(v + 1)λv+1 ∆Tv−1 −
∆Tn−1
n + 1 v=1
(n + 1)pn
= tn,1 + tn,2 + tn,3 + tn,4 ,
say.
Since
|tn,1 + tn,2 + tn,3 + tn,4 |k ≤ 4k (|tn,1 |k + |tn,2 |k + | |tn,3 |k + |t4|k ),
J. Inequal. Pure and Appl. Math., 4(4) Art. 66, 2003
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I NCLUSION T HEOREMS FOR A BSOLUTE S UMMABILITY M ETHODS
7
to complete the proof of Theorem 2.2, it is enough to show that
∞
X
nδk−1 |tn,r |k < ∞
f or
r = 1, 2, 3, 4.
n=1
Now, when k > 1 applying Hölder’s inequality with indices k and k 0 , where
have that
m+1
X
n−1
X
Pv
m+1
X
1
nδk−1 |tn,1 |k ≤
nδk−1 k
n
n=2
n=2
≤
m+1
X
n=2
≤
n=2
n2−δk
v=1
n2−δk
v=1
m X
v=1
= O(1)
= O(1)
Pv
pv
Pv
pv
k
k
= O(1)
as
k
|∆λv | v k |∆Tv−1 |
= 1, we
k
k
m+1
X
n=v+1
|∆λv |k v k |∆Tv−1 |k
pv
!k−1
n−1
1X
1
n v=1
|∆λv |k v k |∆Tv−1 |k
k
pv
v=1
δk+k−1
m X
Pv
v=1
k
|∆λv | v |∆Tv−1 |
k
m X
Pv
1
k0
v |∆λv | |∆Tv−1 |
pv
n−1 X
1
= O(1)
pv
+
!k
k
n−1 X
Pv
1
m+1
X
v=1
1
k
1
n2−δk
1
v 1−δk
|∆Tv−1 |k
m → ∞,
by virtue of the hypotheses of Theorem 2.2.
Again using Hölder’s inequality,
m+1
X
k
nδk−1 |tn,2 | ≤
n=2
m+1
X
!k
Pv
|λv+1 |
|∆Tv−1 |
pv
v=1
n−1
X
nδk−1−k
n=2
≤
m+1
X
1
n2−δk
k
n−1 X
Pv
|λv+1 |k |∆Tv−1 |k
!k−1
n−1
1X
1
n v=1
pv
k
m+1
X
Pv
1
k
k
= O(1)
|λv+1 | |∆Tv−1 |
2−δk
pv
n
v=1
n=v+1
m
k
X
Pv
1
= O(1)
|λv+1 |k |∆Tv−1 |k 1−δk
pv
v
v=1
m
X
Pv δk+k−1
= O(1)
)
|∆Tv−1 |k
p
v
v=1
n=2
v=1
m X
= O(1)
J. Inequal. Pure and Appl. Math., 4(4) Art. 66, 2003
as
m → ∞,
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8
H. S. Ö ZARSLAN
Also, we have that
m+1
X
nδk−1 |tn,3 |k ≤
n=2
m+1
X
n−1
X
nδk−1−k
n=2
!k
(v + 1) |λv+1 | |∆Tv−1 |
v=1
= O(1)
m+1
X
n−1
X
nδk−1−k
n=2
= O(1)
= O(1)
= O(1)
= O(1)
m
X
v=1
m
X
n−1
X
1
n2−δk
m+1
X
n=2
v |λv+1 | |∆Tv−1 |
v=1
m+1
X
n=2
!k
v=1
n−1
X
1
n2−δk
v k |λv+1 |k |∆Tv−1 |k
!k−1
n−1
1X
1
n v=1
v k |λv+1 |k |∆Tv−1 |k
v=1
v k |λv+1 |k |∆Tv−1 |k
m+1
X
n=v+1
1
n2−δk
v δk+k−1 |∆Tv−1 |k
v=1
= O(1)
δk+k−1
m X
Pv
pv
v=1
|∆Tv−1 |k
as m → ∞,
= O(1)
by virtue of the hypotheses of Theorem 2.2.
Finally, we have that
m
X
n
δk−1
k
|tn,4 | = O(1)
n=1
= O(1)
k
m X
Pn
nδk−1 |λn |k |∆Tn−1 |k
pn
n=1
δk+k−1
m
X
Pn
n=1
= O(1)
as
pn
|∆Tn−1 |k
m → ∞,
by virtue of the hypotheses of Theorem 2.2.
Therefore, we get that
m
X
nδk−1 |tn,r |k = O(1)
as m → ∞,
for
r = 1, 2, 3, 4.
n=1
This completes the proof of Theorem 2.2.
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[3] H. BOR, On the local property of N̄ , pn ; δ k summability of factored Fourier series, J. Math. Anal.
Appl., 179 (1993), 646–649.
[4] T.M. FLETT, On an extension of absolute summability and some theorems of Littlewood and Paley,
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I NCLUSION T HEOREMS FOR A BSOLUTE S UMMABILITY M ETHODS
9
[5] T.M. FLETT, Some more theorems concerning the absolute summability of Fourier series, Proc.
London Math. Soc., 8 (1958), 357–387.
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