Download FACTORS FOR ABSOLUTE WEIGHTED ARITHMETIC MEAN

International Journal of Analysis and Applications
ISSN 2291-8639
Volume 14, Number 2 (2017), 175-179
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FACTORS FOR ABSOLUTE WEIGHTED ARITHMETIC MEAN SUMMABILITY
OF INFINITE SERIES
HÜSEYİN BOR∗
Abstract. In this paper, we proved a general theorem dealing with absolute weighted arithmetic
mean summability factors of infinite series under weaker conditions. We have also obtained some
known results.
1. Introduction
Let
an be a given infinite series with partial sums (sn ). We denote by uα
n the nth Cesàro mean
of order α, with α > −1, of the sequence (sn ), that is (see [4])
P
uα
n =
n
1 X α−1
An−v sv ,
Aα
n v=0
(1.1)
where
(α + 1)(α + 2)....(α + n)
= O(nα ), Aα
−n = 0 for
n!
P
A series
an is said to be summable | C, α |k , k ≥ 1, if (see [5])
Aα
n =
∞
X
n > 0.
α
k
nk−1 | uα
n − un−1 | < ∞.
(1.2)
(1.3)
n=1
If we take α=1, P
then we obtain | C, 1 |k summability. Let (pn ) be a sequence of positive numbers
n
such that Pn = v=0 pv → ∞ as n → ∞, (P−i = p−i = 0, i ≥ 1). The sequence-to-sequence
transformation
n
1 X
pv sv
(1.4)
wn =
Pn v=0
defines the sequence (wn ) of the weighted arithmetic mean or simply the P
(N̄ , pn ) mean of the sequence
(sn ), generated by the sequence of coefficients (pn ) (see [6]). The series
an is said to be summable
| N̄ , pn |k , k ≥ 1, if (see [1])
∞
X
(Pn /pn )k−1 | wn − wn−1 |k < ∞.
(1.5)
n=1
If we take pn = 1 for all values of n, then we obtain | C, 1 |k summability. Also if we take k = 1, then
we obtain | N̄ , pn | summability (see [11]). For any sequence (λn ) we write that ∆λn = λn − λn+1 .
2. Known Result
The following theorem is known dealing with | N̄ , pn |k summability factors of infinite series.
2010 Mathematics Subject Classification. 26D15, 40D15, 40F05, 40G99.
Key words and phrases. weighted arithmetic mean; absolute summability; summability factors; infinite series; Hölder
inequality; Minkowski inequality.
c
2017
Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
175
176
BOR
Theorem 2.1. [2] Let (Xn ) be a positive non-decreasing sequence and suppose that there exists
sequences (βn ) and (λn ) such that
| ∆λn |≤ βn ,
(2.1)
βn → 0 as n → ∞,
∞
X
n | ∆βn | Xn < ∞,
(2.2)
(2.3)
n=1
| λn | Xn = O(1).
(2.4)
If
m
X
| sn |k
= O(Xm )
n
n=1
m → ∞,
as
(2.5)
and (pn ) is a sequence such that
then the series
P∞
Pn λn
n=1 an npn
Pn = O(npn ),
(2.6)
Pn ∆pn = O(pn pn+1 ),
(2.7)
is summable | N̄ , pn |k , k ≥ 1.
Remark 2.1. It should be noted that, under the conditions on the sequence (λn ) we have that (λn ) is
bounded and ∆λn = O(1/n) [2].
3. 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 positive non-decreasing sequence. If the sequences (Xn ), (βn ), (λn ), and
(pn ) satisfy the conditions (2.1)-(2.4), (2.6)-(2.7), and
m
X
| sn |k
= O(Xm )
nXnk−1
n=1
then the series
P∞
n=1
m → ∞,
as
(3.1)
n λn
is summable | N̄ , pn |k , k ≥ 1.
an Pnp
n
Remark 3.1. It should be noted that condition (3.1) is the same as condition (2.5) when k=1. When
k > 1, condition (3.1) is weaker than condition (2.5) but the converse is not true. As in [10], we can
show that if (2.5) is satisfied, then we get
m
m
X
X
| sn |k
1
| sn |k
=
O(
)
= O(Xm )
k−1
k−1
n
nXn
X1
n=1
n=1
as
m → ∞.
To show that the converse is false when k > 1, as in [3], the following example is sufficient. We can
take Xn = nδ , 0 < δ < 1, and then construct a sequence (un ) such that
un =
|sn |k
= Xn − Xn−1 ,
nXn k−1
hence
m
X
|sn |k
= Xm = mδ ,
k−1
nX
n
n=1
and so
m
X
|sn |k
n
n=1
=
m
X
(Xn − Xn−1 )Xnk−1 =
n=1
m
X
≥ δ
n=1
m
X
(nδ − (n − 1)δ )nδ(k−1)
n=1
nδ−1 nδ(k−1) = δ
m
X
n=1
nδk−1 ∼
mδk
k
as
m → ∞.
ABSOLUTE WEIGHTED ARITHMETIC MEAN SUMMABILITY
177
It follows that
m
1 X |sn |k
→ ∞ as
Xm n=1 n
m→∞
provided k > 1. This shows that (2.5) implies (3.1) but not conversely.
We require the following lemmas for the proof of Theorem 3.1.
Lemma 3.1. [7] Under the conditions on (Xn ), (βn ) and (λn ) as as expressed in the statement of
the theorem, we have the following;
nXn βn = O(1),
∞
X
(3.2)
βn Xn < ∞.
(3.3)
n=1
Lemma 3.2. [9] If the conditions (2.6) and (2.7) are satisfied, then ∆
4. Proof of Theorem 3.1
P∞
Proof. Let (Tn ) be the sequence of (N̄ , pn ) mean of the series n=1
have
Tn =
Pn
npn
an Pn λn
npn .
=O
1
n
.
Then, by definition, we
n
v
n
1 X X ar Pr λr
av Pv λv
1 X
pv
(Pn − Pv−1 )
=
.
Pn v=1 r=1 rpr
Pn v=1
vpv
Then we get that
Tn − Tn−1 =
n
X
Pv−1 Pv av λv
pn
,
Pn Pn−1 v=1
vpv
n ≥ 1,
(P−1 = 0).
By using Abel’s transformation, we have that
n−1
X
Pv−1 Pv λv
pn
λn sn
sv ∆
Tn − Tn−1 =
+
Pn Pn−1 v=1
vpv
n
=
n−1
n−1
n−1
X Pv+1 Pv ∆λv
X
X
pn
Pv
sn λn
pn
pn
1
+
sv
Pv sv λv ∆
sv Pv λv
+
−
n
Pn Pn−1 v=1 (v + 1)pv+1
Pn Pn−1 v 1
vpv
Pn Pn−1 v=1
v
=
= Tn,1 + Tn,2 + Tn,3 + Tn,4 .
To complete the proof of the Theorem 3.1, by Minkowski’s inequality, it is sufficient to show that
k−1
∞ X
Pn
| Tn,r |k < ∞, for r = 1, 2, 3, 4.
(4.1)
p
n
n=1
Applying Abel’s transformation, we have that
k−1
k−1
k−1
m m m
X
X
X
Pn
Pn
| sn |k
| sn |k
1
| Tn,1 |k =
| λn |k−1 | λn |
= O(1)
| λn |
pn
npn
n
n
Xn
n=1
n=1
n=1
= O(1)
m−1
X
n=1
= O(1)
m−1
X
n=1
∆ | λn |
n
m
m−1
X
X
X
| sv |k
| sn |k
+
O(1)
|
λ
|
=
O(1)
| ∆λn | Xn + O(1) | λm | Xm
m
k−1
k−1
vXv
nXn
v=1
n=1
n=1
βn Xn + O(1) | λm | Xm = O(1),
as
m → ∞,
178
BOR
by the hypotheses of Theorem 3.1 and Lemma 3.1. Now, by using (2.6) and applying Hölder’s inequality, we obtain that
(n−1
)k
m+1
m+1
n−1
m+1
X Pn k−1
X
X
X Pv
X
pn
pn
k
k
| Tn,2 | = O(1)
|
| sv | pv | ∆λv |
Pv sv ∆λv | = O(1)
pn
p
P Pk
P Pk
n=2
n=2 n n−1 v=1
v=1 v
n=2 n n−1
!k−1
n−1
n−1
m+1
X Pv k
X
X
1
pn
= O(1)
| sv |k pv βv k ×
pv
P
P
p
Pn−1 v=1
n
n−1
v
v=1
n=2
k
m+1
m X
X
pn
Pv
| sv |k pv βv k
= O(1)
p
P
Pn−1
v
n
n=v+1
v=1
m
m
k−1
X
X
Pv
= O(1)
βv k−1 βv | sv |k = O(1)
(vβv )k−1 βv | sv |k
p
v
v=1
v=1
k−1
m m
X
X
1
| sv |k
βv | sv |k = O(1)
= O(1)
vβv
Xv
vXv k−1
v=1
v=1
= O(1)
m−1
X
∆(vβv )
v=1
= O(1)
m−1
X
v
m−1
m
X
X
X
| sr |k
| sv |k
| ∆(vβv ) | Xv + O(1)mβm Xm
+
O(1)mβ
=
O(1)
m
rXr k−1
vXv k−1
r=1
v=1
v=1
| (v + 1)∆βv − βv | Xv + O(1)mβm Xm
v=1
= O(1)
m−1
X
v | ∆βv | Xv + O(1)
v=1
m−1
X
Xv βv + O(1)mβm Xm = O(1),
v=1
as m → ∞, by the hypotheses of the Theorem 3.1 and Lemma 3.1. Again, as in Tn,1 , we have that
m+1
X
n−1
X
Pv
pn
Pv sv λv ∆
|k
P
P
vp
n n−1 v 1
v
n=2
n=2
=
)
)k
(
(
k
m+1
m+1
n−1
n−1
X
X
X
X Pv pn
pn
1
1
= O(1)
= O(1)
Pv | sv || λv |
pv | sv || λv |
v
pv
v
P Pk
P Pk
n=2 n n−1
n=2 n n−1
v=1
v=1
)
(
k−1
m+1
n−1
n−1
X
X Pv k
pn
1 X
= O(1)
pv
pv | sv |k | λv |k ×
P P
vpv
Pn−1 v=1
n=2 n n−1 v=1
k
k
m m+1
m X
X
X
Pv
pn
Pv
1 v
k
k
= O(1)
| sv | pv | λv |
= O(1)
pv | sv |k | λv |k
.
vpv
P P
vpv
Pv v
v=1
n=v+1 n n−1
v=1
k−1
k−1
m m m
X
X
X
Pv
| sv |k
1
| sv |k
| sv |k
= O(1)
| λv |k−1 | λv |
= O(1)
| λv |
= O(1)
| λv |
vpv
v
Xv
v
vXv k−1
v=1
v=1
v=1
Pn
pn
= O(1)
k−1
m−1
X
| Tn,3 |k =
m+1
X
Pn
pn
k−1
|
Xv βv + O(1)Xm | λm |= O(1),
as
m → ∞,
v=1
by the hypotheses of the Theorem 3.1, Lemma 3.1 and Lemma 3.2. Finally, using Hölder’s inequality,
as in Tn,3 , we have get
m+1
X
n=2
=
Pn
pn
m+1
X
n=2
k−1
| Tn,4 |k =
m+1
X
n=2
n−1
X Pv
pn
|
sv λv |k
k
v
Pn Pn−1 v=1
k
n−1
m+1
n−1
X Pv
X
X
pn
pn
Pv
k
k
|
sv
pv λv | ≤
| sv |
pv | λv |k ×
k
vp
P
P
vp
Pn Pn−1
v
n
n−1
v
v=1
n=2
v=1
1
n−1
X
Pn−1
v=1
!k−1
pv
ABSOLUTE WEIGHTED ARITHMETIC MEAN SUMMABILITY
179
k
k−1
m m X
X
1 v
Pv
| sv |k
Pv
| sv |k pv | λv |k
| λv |k−1 | λv |
. = O(1)
vpv
Pv v
vpv
v
v=1
v=1
m
m
k−1
X
X
1
| sv |k
| sv |k
= O(1)
| λv |
= O(1)
| λv |
Xv
v
vXv k−1
v=1
v=1
= O(1)
= O(1)
m−1
X
Xv βv + O(1)Xm | λm |= O(1),
as
m → ∞.
v=1
This completes the proof of Theorem 3.1.
5. Conclusions
It should be noted that if we take pn = 1 for all n, then we obtain a known result of Mishra and
Srivastava dealing with | C, 1 |k summability factors of infinite series (see [8]). Also, if we set k = 1,
then we have a known result of Mishra and Srivastava concerning the | N̄ , pn | summability factors of
infinite series (see [9]).
References
[1] H. Bor, On two summability methods, Math. Proc. Camb. Philos Soc. 97 (1985), 147-149.
[2] H. Bor, A note on | N̄ , pn |k summability factors of infinite series, Indian J. Pure Appl. Math. 18 (1987), 330-336.
[3] H. Bor, Quasi-monotone and almost increasing sequences and their new applications, Abstr. Appl. Anal. 2012, Art.
ID 793548, 6 pp.
[4] E. Cesàro, Sur la multiplication des séries, Bull. Sci. Math. 14 (1890), 114-120.
[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] G. H. Hardy, Divergent Series, Clarendon Press, Oxford, (1949).
[7] K. N. Mishra, On the absolute Nörlund summability factors of infinite series, Indian J. Pure Appl. Math. 14 (1983),
40-43.
[8] K. N. Mishra and R. S. L. Srivastava, On the absolute Cesàro summability factors of infinite series, Portugal. Math.
42 (1983/84), 53-61.
[9] K. N. Mishra and R. S. L. Srivastava, On | N̄ , pn | summability factors of infinite series, Indian J. Pure Appl. Math.
15 (1984), 651-656.
[10] W. T. Sulaiman, A note on |A|k summability factors of infinite series, Appl. Math. Comput. 216 (2010), 2645-2648.
[11] G. Sunouchi, Notes on Fourier analysis. XVIII. Absolute summability of series with constant terms, Tôhoku Math.
J. (2), 1 (1949), 57-65.
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∗ Corresponding
author: [email protected]