Filomat 28:6 (2014), 1205–1209
DOI 10.2298/FIL1406205B
Published by Faculty of Sciences and Mathematics,
University of Niš, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
A New Factor Theorem for Generalized Cesàro Summability
Hüseyin Bora
a P.
O. Box 121, TR-06502 Bahçelievler, Ankara, Turkey
Abstract. In [6], we proved a theorem dealing with an application of quasi-f-power increasing sequences.
In this paper, we prove that theorem under less and weaker conditions. This theorem also includes several
new results.
1. Introduction
A positive X = (Xn ) is said to be a quasi-f-power increasing sequence if there exists a constant K =
K(X, f ) ≥ 1 such that K fn Xn ≥ fm Xm for all n ≥ m ≥ 1, where f = ( fn ) = {nσ (log n)η , η ≥ 0, 0 < σ < 1}
(see [13]). If we take η=0, then we get a quasi-σ-power increasing
sequence
P (see [12]). We write BVO =
BV ∩ CO , where CO = { x = (xk ) ∈ Ω : limk |xk | = 0 }, BV=
x
=
(x
)
∈
Ω
:
k
k |xk − xk+1 | < ∞ and Ω being
P
the space of all real or complex-valued sequences. Let an be a given infinite series with partial sums (sn ).
α,δ
We denote by uα,δ
n and tn the nth Cesàro means of order (α, δ), with α + δ > −1, of the sequence (sn ) and
(nan ), respectively, that is (see [8])
uα,δ
n =
tα,δ
n =
1
n
X
Anα+δ
v=0
1
n
X
Aα+δ
n
v=1
δ
Aα−1
n−v Av sv
(1)
δ
Aα−1
n−v Av vav ,
(2)
where
Aα+δ
= O(nα+δ ),
n
Aα+δ
f or n > 0.
(3)
−n = 0
P
Let (ϕn ) be a sequence of complex numbers. The series an is said to be summable ϕ − | C, α, δ |k , k ≥ 1 and
α + δ > −1, if (see [4],[9])
∞
X
n=1
α + δ > −1,
α,δ
k
| ϕn (uα,δ
n − un−1 ) | =
∞
X
Aα+δ
=1
0
and
k
n−k | ϕn tα,δ
n | < ∞.
(4)
n=1
2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G05, 46A45
Keywords. Cesàro mean; absolute summability; increasing sequences; sequence spaces; infinite series; Hölder inequality;
Minkowski inequality.
Received: 18 May 2013; Accepted: 22 November 2013
Communicated by Dragan S. Djordjevic
Email address: [email protected] (Hüseyin Bor)
Hüseyin Bor / Filomat 28:6 (2014), 1205–1209
1206
1
In the special case when ϕn = n1− k , ϕ − | C, α, δ |k summability is the same as | C, α, δ |k summability (see
1
[9]). Also, if we set ϕn = nγ+1− k , then ϕ − | C, α, δ |k summability reduces to | C, α, δ; γ |k summability (see
1
[4]). If we take δ = 0, then we have ϕ − | C, α |k summability (see [1]). Furthermore, if we take ϕn = n1− k and
1
β = 0, then we get | C, α |k summability (see [10]). Finally, if we take ϕn = nγ+1− k and δ = 0, then we obtain
| C, α; γ |k summability (see [11]).
2. The known result. The following theorem is known.
Theorem A ([6]). Let (λn ) ∈ BVO and let (Xn ) be a quasi-f-power increasing sequence for some σ (0 < σ < 1).
Suppose also that there exist sequences (βn ) and (λn ) such that
| ∆λn |≤ βn ,
βn → 0
∞
X
as
(5)
n → ∞,
(6)
n | ∆βn | Xn < ∞,
(7)
n=1
| λn | Xn = O(1)
as
n → ∞.
(8)
If there exists an > 0 such that the sequence (n−k | ϕn |k ) is non-increasing and if the sequence (θα,δ
n ) is
defined by
(
tα,δ
α = 1, δ > −1
α,δ
n ,
θn =
(9)
max1≤v≤n tα,δ
,
0
<
α < 1, δ > −1
v
satisfies the condition
m
k
X
(| ϕn | θα,δ
n )
= O(Xm )
nk
n=1
as
m → ∞,
(10)
P
then the series an λn is summable ϕ− | C, α, δ |k , k ≥ 1, 0 < α ≤ 1 , δ > −1,η ≥ 0 and (α + δ)k + > 1.
Remark 2.2 It should ne noted that in the statement of Theorem 2.1, a different notation has been used for
the quasi-f-power increasing sequences. If we take η = 0, then we obtain a known theorem (see [5]).
3. The main result
The aim of this paper is to prove Theorem 2.1 under weaker conditions. Now, we shall prove the following
theorem :
Theorem 3.1 Let (Xn ) be a quasi-f-power increasing sequence and the sequences (λn ) and (βn ) such that
conditions (5)-(8) are satisfied. If there exists an > 0 such that the sequence (n−k | ϕn |k ) is non-increasing
and if the condition
m
k
X
(| ϕn | θα,δ
n )
= O(Xm ) as m → ∞,
(11)
nk Xnk−1
n=1
P
satisfies, then the series an λn is summable ϕ− | C, α, δ |k , k ≥ 1, 0 < α ≤ 1, δ > −1, η ≥ 0 and (α + δ−1)k+ >
0.
Remark 3.2 It should be noted that condition (11) is the same as condition (10) when k=1. When k > 1,
condition (11) is weaker than condition (10), but the converse is not true. As in [14] we can show that if (10)
is satisfied, then we get that
m
k
X
(| ϕn | θα,δ
n )
n=1
nk Xnk−1
= O(
m
α,δ
1 X (| ϕn | θn )k
)
= O(Xm ).
nk
X1k−1 n=1
If (11) is satisfied, then for k > 1 we obtain that
m
m
m
α,δ k
k
k
X
X
X
(| ϕn | θα,δ
(| ϕn | θα,δ
n )
n )
k−1 (| ϕn | θn )
k−1
k
=
X
=
O(X
)
= O(Xm
) , O(Xm ).
m
n
k
k Xk−1
k Xk−1
n
n
n
n
n
n=1
n=1
n=1
Hüseyin Bor / Filomat 28:6 (2014), 1205–1209
1207
Also it should be noted that the condition ”(λn ) ∈ BVO ” has been removed.
We need the following lemmas for the proof of our theorem.
Lemma 3.3 ([3]) If 0 < α ≤ 1, δ > −1 and 1 ≤ v ≤ n, then
|
v
X
δ
Aα−1
n−p Ap ap |≤ max |
1≤m≤v
p=0
m
X
δ
Aα−1
m−p Ap ap | .
(12)
p=0
Lemma 3.4 ([7]) Under the conditions on (Xn ), (βn ) and (λn ) as expressed in the statement of the theorem,
we have the following ;
nXn βn = O(1),
∞
X
(13)
βn Xn < ∞.
(14)
n=1
4. Proof of Theorem 3.1 Let (Tnα,δ ) be the nth (C, α, δ) mean of the sequence
(nan λn ). Then, by (2), we have that
Tnα,δ
=
1
n
X
Aα+δ
n
v=1
δ
Aα−1
n−v Av vav λv .
First applying Abel’s transformation and then use of Lemma 3.3, we have that
Tnα,δ
| Tnα,δ |
=
≤
≤
=
1
n−1
X
Aα+δ
n
v=1
1
Aα+δ
n
∆λv
n−1
X
v
X
δ
Aα−1
n−p Ap pap +
p=1
| ∆λv ||
v=1
v
X
δ
Aα−1
n−p Ap pap | +
p=1
n−1
X
1
Aαv Aδv θα,δ
v
α+δ
An v=1
α,δ
α,δ
Tn,1
+ Tn,2
.
n
λn X α−1 δ
An−v Av vav ,
Aα+δ
n
v=1
n
| λn | X α−1 δ
|
An−v Av vav |
Aα+δ
n
v=1
| ∆λv | + | λn | θα,δ
n
To complete the proof of the theorem, by Minkowski’s inequality, it is sufficient to show that
∞
X
n=1
α,δ k
n−k | ϕn Tn,r
| <∞
f or
r = 1, 2.
Hüseyin Bor / Filomat 28:6 (2014), 1205–1209
1208
Now, when k > 1, applying Hölder’s inequality with indices k and k0 , where
m+2
X
α,δ k
n−k | ϕn Tn,1
|
≤
n=2
m+1
X
−k
n−k (Aα+δ
n )
| ϕn |k {
n=2
≤
m+1
X
−k −(α+δ)k
n n
| ϕn |
k
O(1)
n−1
X
m
X
O(1)
=
O(1)
O(1)
αk
v
k
k
v(α+δ) (θα,δ
v ) (βv )
k
m
X
k k
v(α+δ)k (θα,δ
v ) βv
v=1
=
k
(θα,δ
v ) |
= 1, we get that
m
X


n−1 k−1


X 

∆λv | × 
1




O(1)
k
v=1
m+1
X
n−k | ϕn |k
n(α+δ)k+1−k
n=v+1
m+1
X
n−k | ϕn |k
n1+(α+δ−1)k+
n=v+1
k
k −k
| ϕv |k
v(α+δ)k (θα,δ
v ) (βv ) v
m+1
X
v=1
n=v+1
m
X
Z
k
k −k
v(α+δ)k (θα,δ
| ϕv |k
v ) (βv ) v
m
X
∞
v
v=1
=
1
k0
α,δ
k
Aα+δ
v θv | ∆λv |}
v=1
v=1
=
+
v=1
n=2
=
n−1
X
1
k
1
n1+(α+δ−1)k+
dx
x1+(α+δ−1)k+
k
βv (βv )k−1 (θα,δ
v | ϕv |)
v=1
=
O(1)
m
X
βv
v=1
=
O(1)
m−1
X
O(1)
1
vXv
∆(vβv )
v=1
=
k−1
v
k
X
(| ϕr | θα,δ
r )
r=1
m−1
X
k
(θα,δ
v | ϕv |)
rk Xrk−1
+ O(1)mβm
m
k
X
(| ϕv | θα,δ
v )
v=1
vk Xvk−1
| ∆(vβv ) | Xv + O(1)mβm Xm
v=1
=
O(1)
m−1
X
| (v + 1)∆βv − βv | Xv + O(1)mβm Xm
v=1
=
O(1)
m−1
X
v | ∆βv | Xv + O(1)
v=1
= O(1)
as
m−1
X
v=1
m → ∞,
βv Xv + O(1)mβm Xm
Hüseyin Bor / Filomat 28:6 (2014), 1205–1209
1209
virtue of the hypotheses of the theorem and Lemma 3.4. Finally, we have that
m
X
α,δ k
n−k | ϕn Tn,2
|
=
n=1
m
X
k
| λn | | λn |k−1 n−k (θα,δ
n | ϕn |)
n=1
=
O(1)
m
X
n=1
=
O(1)
m−1
X
n=1
=
O(1)
m−1
X
| λn |
1
Xn
∆ | λn |
k−1
k
n−k (θα,δ
n | ϕn |)
n
k
X
(| ϕr | θα,δ
r )
r=1
rk Xrk−1
+ O(1) | λm |
m
k
X
(| ϕn | θα,δ
n )
n=1
nk Xnk−1
| ∆λn | Xn + O(1) | λm | Xm
n=1
=
O(1)
m−1
X
βn Xn + O(1) | λm | Xm = O(1)
as
m → ∞,
n=1
by virtue of the hypotheses of the theorem and Lemma 3.4. This completes the proof of Theorem 3.1. If
1
we set = 1 and ϕn = n1− k , then we obtain a new result concerning the | C, α, δ |k summability. Also, if we
1
take ϕn = nγ+1− k , then we obtain another new result dealing with the | C, α, δ; γ |k summability factors. If
we take η = 0, then we obtain Theorem A under weaker conditions. If we set η = 0 and δ=0, then we obtain
the result of Bor and Özarslan under weaker conditions (see [2]). Furthermore, if we take = 1, δ = 0 and
1
ϕn = nγ+1− k , then we get the result of Bor under weaker conditions (see [7]).
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[11]
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