Journal of Inequalities in Pure and
Applied Mathematics
INCLUSION THEOREMS FOR ABSOLUTE SUMMABILITY
METHODS
volume 4, issue 4, article 66,
2003.
H. S. ÖZARSLAN
Department of Mathematics,
Erciyes University,
38039 Kayseri,
Turkey.
EMail: seyhan@erciyes.edu.tr
Received 07 April, 2003;
accepted 30 April, 2003.
Communicated by: L. Leindler
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
046-03
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Abstract
In this paper we have proved two theorems concerning an inclusion between
two absolute summability methods by using any absolute summability factor.
2000 Mathematics Subject Classification: 40D15, 40F05, 40G99
Key words: Absolute summability, Summability factors, Infinite series.
Inclusion Theorems for
Absolute Summability Methods
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
6
H. S. Özarslan
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1.
Introduction
P
Let
an be a given infinite series with partial sums (sn ), and rn = nan . By
un and tn we denote the
P 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
[5])
P
(1.3)
Contents
nδk−1 |tn |k < ∞.
If we take δ = 0, then |C, 1; δ|k summability is the same as |C, 1|k summability.
Let (pn ) be a sequence of positive numbers such that
(1.4)
Pn =
v=0
pv → ∞ as n → ∞,
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n=1
n
X
H. S. Özarslan
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an is said to be summable |C, 1; δ|k k ≥ 1 and δ ≥ 0, if (see
∞
X
Inclusion Theorems for
Absolute Summability Methods
(P−i = p−i = 0, i ≥ 1).
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The sequence-to-sequence transformation
n
1 X
Tn =
p v sv
Pn v=0
(1.5)
defines the sequence (Tn ) of the (N̄ , pn ) mean of the sequence (sn ), generated
by the sequence of coefficients (pn ) (see [6]).
P
The series an is said to be summable N̄ , pn k , k ≥ 1, if (see [1])
(1.6)
k−1
∞ X
Pn
n=1
pn
|∆Tn−1 |k < ∞
and it is said to be summable N̄ , pn ; δ k , k ≥ 1 and δ ≥ 0, if (see [3])
(1.7)
δk+k−1
∞ X
Pn
n=1
pn
|∆Tn−1 |k < ∞,
where
(1.8)
∆Tn−1
n
pn X
=−
Pv−1 av ,
Pn Pn−1 v=1
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
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n ≥ 1.
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N̄ , pn ; δ In the special case when δ = 0 (resp.
p
=
1
for
all
values
of
n)
n
k
summability is the 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.
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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
P numbers such that
(1.9) of Theorem 1.1 is satisfied. If the series
an is
condition
summable N̄ , pn k , then it is also summable |C, 1|k .
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
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2.
The Main Result
The
aim of this paper is to generalize the above theorems for |C, 1; δ|k and
N̄ , pn ; δ summabilities, by using a summability factors. Now, we shall prove
k
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
1
Pv
1
Pn
(2.1)
=O
.
pn
Pn−1
pv
Pv
n=v+1
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
(2.3)
λn = O
Pn
npn
k1
.
Theorem 2.2. Let k ≥ 1 and 0 ≤ δk < 1. Let (pn ) be a sequence of positive
P
numbers such that npn = O(PP
an
n ) and satisfies the condition (2.1). Let
be summable N̄ , pn ; δ k . Then an λn is summable |C, 1; δ|k , if (λn ) satisfies
the following conditions:
1−δk
k
npn
(2.4)
n∆λn = O
Pn
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
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λn = O
(2.5)
npn
Pn
k1
.
Remark 2.1. 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
X 1
1
1
pn
=
−
=O
P P
Pn−1 Pn
Pv
n=v+1 n n−1
n=v+1
m+1
X
as
m → ∞,
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
which always holds.
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Proof of Theorem 2.1. Since
Contents
tn =
1
n+1
n
X
vav
v=1
we have that
n+1
an =
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
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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 ,
say.
H. S. Özarslan
Title Page
Contents
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
(2.6)
Inclusion Theorems for
Absolute Summability Methods
∞ X
n=1
Pn
pn
δk+k−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
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1
k
+
1
k0
= 1, we get
m+1
X
n=2
≤
Pn
pn
δk+k−1
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+1
X
n=2
m
X
v=1
m
X
v=1
= O(1)
m
X
|Tn,1 |k
Pn
pn
!k
v+1
Pv
|tv | |∆λv |
v
v=1
n−1
X
1
k
Pn−1
δk−1
n−1
X
1
Pv
|tv | |v∆λv |
pv
vpv
v=1
!k
n−1
X
|tv | |v∆λv | pv
k
Pn−1
Pn
pn
δk−1
Pn
pn
δk−1
1
k
Pn−1
Pn−1
|tv |k |v∆λv |k pv
H. S. Özarslan
Title Page
v=1
m+1
X
|tv |k |v∆λv |k pv
Pn
pn
n=v+1
δk−1
Pv
k
|tv |k |v∆λv |
Inclusion Theorems for
Absolute Summability Methods
v=1
n−1
X
1
!k
δk−1
1
Pn−1
pv
v δk−1 |tv |k = O(1)
1
n−1
X
Pn−1
v=1
!k−1
pv
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as m → ∞.
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v=1
J. Ineq. Pure and Appl. Math. 4(4) Art. 66, 2003
by virtue of the hypotheses of Theorem 2.1.
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Again using Hölder’s inequality,
m+1
X Pn δk+k−1
|Tn,2 |k
pn
n=2
!k−1
m+1
n−1
n−1
X Pn δk−1 1 X
X
1
= O(1)
pv |tv |k |λv |k
pv
p
P
P
n
n−1
n−1
n=2
v=1
v=1
δk−1
m
m+1
X
X
1
Pn
= O(1)
pv |tv |k |λv |k
pn
Pn−1
v=1
n=v+1
δk−1
m X
Pv
= O(1)
|tv |k |λv |k
p
v
v=1
= O(1)
= O(1)
m
X
v=1
m
X
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
Title Page
v
δk−1
k
k
|tv | |λv |
v δk−1 |tv |k = O(1)
Contents
as m → ∞,
v=1
by virtue of the hypotheses of Theorem 2.1.
Also
m+1
X Pn δk+k−1
|Tn,3 |k
p
n
n=2
!k
m+1
n−1
X Pn δk−1 1
X
|λv+1 |
≤
Pv
|tv |
k
pn
v
Pn−1
n=2
v=1
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= O(1)
m+1
X
n=2
= O(1)
= O(1)
= O(1)
= O(1)
m+1
X
n=2
m
X
v=1
m
X
v=1
m
X
Pn
pn
δk−1
Pn
pn
δk−1
k
Pn−1
k
pv |λv+1 | |tv |
n−1
X
1
Pn−1
k
|tv |k
!k
|λv+1 |
|tv |
vpv
v
v=1
n−1
X
1
Pv
pv
pv |λv+1 |k |tv |k
v=1
m+1
X n=v+1
δk−1
Pn
pn
δk−1
1
n
X
Pn−1
v=1
!k−1
pv
1
Pn−1
Inclusion Theorems for
Absolute Summability Methods
|λv+1 |k
H. S. Özarslan
v δk−1 |tv |k = O(1)
as
m → ∞,
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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
pn
pn
n=1
n=1
δk−1
m
X
Pn
= O(1)
|λn |k |tn |k nδk−1
np
n
n=1
= O(1)
m
X
nδk−1 |tn |k = O(1)
n=1
by virtue of the hypotheses of Theorem 2.1.
as
m → ∞,
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This completes the proof of Theorem 2.1.
Proof of Theorem 2.2. Let (Tn ) denotes the (N̄ , pn ) mean of the series
We have
n
n
1 X
1 X
p v sv =
(Pn − Pv−1 )av .
Tn =
Pn v=0
Pn v=0
Since
Tn − Tn−1
n
pn X
=
Pv−1 av ,
Pn Pn−1 v=0
we have that
P
an .
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
−Pn
Pn−2
an =
∆Tn−1 +
∆Tn−2 .
pn
pn−1
(2.7)
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
1 X
Pv
nPn λn
(−v) ∆Tv−1 λv −
=
∆Tn−1
n + 1 v=1
pv
(n + 1)pn
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n
1 X Pv−2
+
v
∆Tv−2 λv
n + 1 v=1 pv−1
J. Ineq. Pure and Appl. Math. 4(4) Art. 66, 2003
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n−1
1 X
Pv
=
(−v) ∆Tv−1 λv
n + 1 v=1
pv
n−1
nPn λn
1 X
Pv−1
+
(v + 1)
∆Tv−1 λv+1 −
∆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−1 } −
∆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 − pv )} −
∆Tn−1
n + 1 v=1 pv
(n + 1)pn
=
1
n+1
n−1
X
v=1
∆Tv−1
{−vλv Pv + (v + 1)λv+1 Pv − (v + 1)λv+1 pv }
pv
nPn λn
∆Tn−1
(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
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
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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 ),
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
1
+ k10 = 1, we have that
k
!k
m+1
m+1
n−1
X
X
X
1
P
v
nδk−1 |tn,1 |k ≤
nδk−1 k
v |∆λv | |∆Tv−1 |
n
p
n=2
n=2
v=1 v
!k−1
k
n−1 n−1
m+1
X 1 X
Pv
1X
k k
k
≤
|∆λv | v |∆Tv−1 |
1
n2−δk v=1 pv
n v=1
n=2
k
m+1
n−1 X 1 X
Pv
≤
|∆λv |k v k |∆Tv−1 |k
2−δk
n
pv
n=2
v=1
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
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= O(1)
= O(1)
= O(1)
k
m X
Pv
pv
v=1
k
m X
Pv
k
m+1
X
k
|∆λv | v |∆Tv−1 |
n=v+1
|∆λv |k v k |∆Tv−1 |k
pv
v=1
δk+k−1
m
X
Pv
v=1
k
pv
1
n2−δk
1
v 1−δk
|∆Tv−1 |k = O(1)
as
m → ∞,
Inclusion Theorems for
Absolute Summability Methods
by virtue of the hypotheses of Theorem 2.2.
Again using Hölder’s inequality,
m+1
X
nδk−1 |tn,2 |k ≤
n=2
m+1
X
nδk−1−k
n=2
≤
m+1
X
n−1
X
|λv+1 |
v=1
1
H. S. Özarslan
!k
k
n−1 X
Pv
Pv
|∆Tv−1 |
pv
Title Page
n−1
|λv+1 |k |∆Tv−1 |k
1X
1
n v=1
n2−δk v=1 pv
k
m m+1
X
X
Pv
1
= O(1)
|λv+1 |k |∆Tv−1 |k
pv
n2−δk
v=1
n=v+1
k
m X
1
Pv
|λv+1 |k |∆Tv−1 |k 1−δk
= O(1)
pv
v
v=1
m X
Pv δk+k−1
= O(1)
)
|∆Tv−1 |k
pv
v=1
n=2
= O(1)
as m → ∞,
!k−1
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Also, we have that
m+1
X
n=2
n
δk−1
k
|tn,3 | ≤
m+1
X
n
n−1
X
δk−1−k
n=2
= O(1)
!k
(v + 1) |λv+1 | |∆Tv−1 |
v=1
m+1
X
nδk−1−k
n=2
= O(1)
= O(1)
= O(1)
= O(1)
= O(1)
n2−δk
1
n2−δk
v |λv+1 | |∆Tv−1 |
v=1
n−1
X
k
k
v k |λv+1 | |∆Tv−1 |
v=1
Title Page
m+1
X
v δk+k−1 |∆Tv−1 |k
as
Pv
pv
Inclusion Theorems for
Absolute Summability Methods
v k |λv+1 |k |∆Tv−1 |k
n=v+1
v=1
m X
!k−1
n−1
1X
1
n v=1
H. S. Özarslan
v k |λv+1 |k |∆Tv−1 |k
v=1
= O(1)
n−1
X
1
n=2
m+1
X
v=1
m
X
!k
v=1
m+1
X
n=2
m
X
n−1
X
δk+k−1
m → ∞,
1
n2−δk
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|∆Tv−1 |
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by virtue of the hypotheses of Theorem 2.2.
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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
pn
|∆Tn−1 |k
as m → ∞,
= O(1)
Inclusion Theorems for
Absolute Summability Methods
by virtue of the hypotheses of Theorem 2.2.
Therefore, we get that
m
X
nδk−1 |tn,r |k = O(1)
as
m → ∞,
n=1
This completes the proof of Theorem 2.2.
H. S. Özarslan
for
r = 1, 2, 3, 4.
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References
[1] H. BOR, On two summability methods, Math. Proc. Cambridge Phil. Soc.,
97 (1985), 147–149.
[2] H. BOR, A note on two summability methods, Proc. Amer. Math. Soc., 98
(1986), 81–84.
[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, Proc. London Math. Soc., 7 (1957), 113–141.
[5] T.M. FLETT, Some more theorems concerning the absolute summability of
Fourier series, Proc. London Math. Soc., 8 (1958), 357–387.
[6] G.H. HARDY, Divergent Series, Oxford University Press., Oxford, (1949).
[7] E. KOGBETLIANTZ, Sur les series absolument sommables par la methode
des moyannes aritmetiques, Bull. Sci. Math., 49(2) (1925), 234–256.
Inclusion Theorems for
Absolute Summability Methods
H. S. Özarslan
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