Bulletin of Mathematical analysis and Applications
ISSN: 1072-6691. URL: http://www.bmathaa.org
Volume 3, Issue 1, Pages 102-107.
A new factor theorem for absolute Cesàro
summability
(communicated by Naim Braha)
∗
HÜSEYİN BOR
Abstract
In this paper, a known theorem dealing with | C, α; δ |k summability factors has been generalized for | C, α, β ; δ |k summability factors.Our theorem also includes some known results.
1
Introduction
P
Let
an be a given infinite series with partial sums (sn ). We denote by uα,β
n
α,β
and tn the nth Cesàro means of order (α, β), with α + β > −1, of the sequence
(sn ) and (nan ), respectively, i.e., (see [3])
uα,β
=
n
tα,β
=
n
1
n
X
Anα+β
v=0
1
n
X
Anα+β
v=1
α−1 β
An−v
Av sv
(1)
α−1 β
An−v
Av vav ,
(2)
where
Aα+β
= O(nα+β ),
n
The series
P
Aα+β
= 1 and
0
α+β
A−n
= 0 f or
n > 0.
(3)
an is said to be summable | C, α, β |k , k ≥ 1 , if (see [5])
∞
X
α,β k
nk−1 | uα,β
n − un−1 | < ∞.
n=1
∗ 2010 Mathematics Subject Classifications: 40D15, 40F05, 40G05,40G99.
Key words: Absolute Cesàro summability, infinite series, summability factors.
c
2008
Universiteti i Prishtines, Prishtine, Kosovë.
Submitted December 22, 2010. Published February 22, 2011.
102
(4)
HÜSEYİN BOR
103
α,β
Since tα,β
= n(uα,β
n
n − un−1 ) (see [5]), condition (4) can also be written as
∞
X
1 α,β k
|t
| < ∞.
n n
n=1
The series
P
(5)
an is summable | C, α, β ; δ |k , k ≥ 1 and δ ≥ 0 , if (see [2])
∞
X
α,β k
nδk+k−1 | uα,β
n − un−1 | =
n=1
∞
X
nδk−1 | tα,β
|k < ∞.
n
(6)
n=1
If we take δ = 0 , then | C, α, β ; δ |k summability reduces to | C, α, β |k summability.Also if we take β = 0 , then we get | C, α; δ |k summability (see [7]).
Furthermore, if we take β = 0 and δ = 0 , then | C, α, β ; δ |k summability
reduces to | C, α |k summability (see [6]). It should be noted that obviously
(C, α, 0) mean is the same as (C, α) mean. A sequence (λn ) is said to be convex
if ∆2 λn ≥ 0, where ∆2 λn = ∆λn − ∆λn+1 and ∆λn = λn − λn+1 . In [8] Lal and
Singh have proved the following theorem dealing with | C, α; δ |k summability
factors of infinite series.
P
Theorem A. If (λn ) is a convex sequence such that n−1 λn is convergent and
the sequence (θnα ) defined by
θnα =| tα
n |,
α=1
θnα = max | tα
v |, 0 < α < 1
1≤v≤n
(7)
(8)
satisfies the condition
(nδ θnα )k = O{(log n)p+k−1 } (C, 1),
P
then the series (log n)−p−k+1 an λn is summable | C, α; δ |k for 0 < α ≤ 1, p ≥
0, k ≥ 1, δ ≥ 0 and δk < α.
2
The Main Result
The aim of this paper is to generalize Theorem A for | C, α, β ; δ |k summability.
We shall prove the following theorem.
P −1
Theorem . If (λn ) is a convex sequence such that
n λn is convergent and
α,β
the sequence (θn ) defined by
θnα,β =| tα,β
|,
n
α = 1 , β > −1
θnα,β = max | tα,β
|, 0 < α < 1 , β > −1
v
1≤v≤n
(9)
(10)
104
A new factor theorem for absolute Cesàro summability
satisfies the condition
(nδ θnα,β )k = O{(log n)p+k−1 } (C, 1),
P
then the series (log(n + 1))−p+k+1 an λn is summable | C, α, β ; δ |k for 0 <
α ≤ 1, β > −1, k ≥ 1 , δ ≥ 0, p ≥ 0 and α + β − δ > 0 .
We need the following lemmas for the proof of our theorem.
P −1
Lemma 1 ([4]). If (λn ) is a convex sequence such that the series
n λn
is convergent, then (λn ) is non-negative and non-increasing, n∆λn = O(1) and
λn log n = o(1) , as n → ∞.
Lemma 2 ([1]). If 0 < α ≤ 1, β > −1 and 1 ≤ v ≤ n, then
|
v
X
α−1 β
An−p
Ap ap
|≤ max |
1≤m≤v
p=0
m
X
β
Aα−1
m−p Ap ap | .
(11)
p=0
Lemma 3 ([10]). If ((log (n + 1))p+k−1 Xn ) satisfies the same conditions as
(λn ) in Lemma 1, then
n (log (n + 1))p+k−1 ∆Xn = O(1)
, as n → ∞
and
m
X
n (log (n + 1))p+k−1 ∆2 Xn = O(1)
, as m → ∞.
n=1
Lemma 4 ([8]). If (λn ) is a convex sequence such that the series
convergent, then for p ≥ 0 and k ≥ 1
m
X
∆(λn )k
= O(1),
(log (n + 1))p(k+1)+(k−1)2
n=1
3
P
n−1 λn is
as m → ∞.
Proof of the Theorem
We write
Xn =
λn
.
(log (n + 1))p+k−1
Let (Tnα,β ) be the n-th (C, α, β) mean of the sequence (nan Xn ). Then, by (2),
we have
Tnα,β
=
1
n
X
Aα+β
n
v=1
α−1 β
An−v
Av vav Xv .
First applying Abel’s transformation and then using Lemma 2, we have that
Tnα,β
=
1
Aα+β
n
n−1
X
v=1
∆Xv
v
X
i=1
β
Aα−1
n−i Ai iai +
n
Xn X
Anα+β
v=1
α−1 β
An−v
Av vav ,
HÜSEYİN BOR
| Tnα,β |
≤
≤
=
n−1
X
1
Aα+β
n
∆Xv |
v=1
105
v
X
β
Aα−1
n−i Ai iai | +
i=1
n−1
X
1
β α,β
Aα
v Av θv ∆Xv
α+β
An
v=1
α,β
α,β
Tn,1
+ Tn,2
, say.
Xn
Aα+β
n
n
X
|
α−1 β
An−v
Av vav |
v=1
+ Xn θnα,β
Since
α,β k
α,β k
α,β k
α,β
| ≤ 2k (| Tn,1
| + | Tn,2
| ),
+ Tn,2
| Tn,1
in order to complete the proof of the theorem, by (6), it is sufficient to show
that
∞
X
α,β k
nδk−1 | Tn,r
| < ∞ , f or
r = 1, 2.
n=1
Whenever k > 1, we can apply Hölder’s inequality with indices k and k 0 , where
1
1
k + k0 = 1, we get that
m+1
X
α,β k
nδk−1 | Tn,1
|
≤
n=2
m+1
X
nδk−1 |
n=2
=
O(1)
m+1
X
n=2
=
O(1)
m
X
1
n−1
X
Aα+β
n
v=1
β α,β
k
Aα
v Av θv ∆Xv |
n−1
X
1
n
{
1+(α+β−δ)k
v αk v βk ∆Xv (θvα,β )k } × {
v=1
v (α+β)k ∆Xv (θvα,β )k
v=1
=
O(1)
m
X
v
∆Xv (θvα,β )k
=
O(1)
Z
v
v=1
m
X
∆Xv }k−1
v=1
m+1
X
n=v+1
(α+β)k
n−1
X
∞
1
n1+(α+β−δ)k
dx
x1+(α+β−δ)k
∆Xv (v δ θvα,β )k
v=1
=
O(1)
m−1
X
∆(∆Xv )
v=1
+
v
X
(pδ θpα,β )k
p=1
O(1)∆Xm
m
X
(v δ θvα,β )k
v=1
=
O(1)
=
O(1)
m−1
X
v(log (v + 1))p+k−1 ∆2 Xv + O(m(log (m + 1))p+k−1 ∆Xm )
v=1
as
m → ∞,
106
A new factor theorem for absolute Cesàro summability
by the application of Lemma 3. Similarly, we have that
m
X
nδk−1 | Xn θnα,β |k
= O(1)
n=1
= O(1)
m
X
Xnk δ α,β k
(n θn )
n
n=1
m−1
X
∆(n−1 Xnk )
n=1
+ O(1)
= O(1)
k
Xm
m
m−1
X
n
X
(v δ θvα,β )k
v=1
m
X
(v δ θvα,β )k
v=1
k
n(log (n + 1))p+k−1 ∆(n−1 Xnk ) + O(Xm
(log (m + 1))p+k−1 )
n=1
= O(1)
m−1
X
n−1 Xnk (log (n + 1))p+k−1 + O(1)
n=1
m−1
X
n=1
(log (n + 1))p+k−1 ∆Xnk
n=1
+ O(1)((log (m + 1))
= O(1)
m−1
X
p+k−1
k
Xm
)
(λn log (n + 1))k
(n + 1)(log (n + 1))1+p(k−1)+k(k−1)
m−1
X
∆λkn
(log (n + 1))p(k−1)+(k−1)2
n=1
(λm log (m + 1))k
+ O
(log (m + 1))p(k−1)+k(k−1)+1
= O(1) as m → ∞,
+ O(1)
by the application of Lemma 4 and λn log n = O(1). Therefore, by (6), we get
that
∞
X
α,β k
nδk−1 | Tn,r
| < ∞ , f or
r = 1, 2.
n=1
This completes the proof of the theorem. It should be noted that if we take
β = 0, then we obtain Theorem A. This theorem also includes as particular
cases the results of Pati [9] and Prasad and Bhatt [10].
References
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HÜSEYİN BOR
107
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HÜSEYİN BOR
P.O.Box 121 , 06502 Bahçelievler,
Ankara, Turkey
e-mail: hbor33@gmail.com