A GENERAL NOTE ON INCREASING SEQUENCES
HÜSEYİN BOR
Department of Mathematics
Erciyes University
38039 Kayseri, TURKEY
EMail: bor@erciyes.edu.tr
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
Title Page
Received:
23 December, 2006
Accepted:
08 August, 2007
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
40D15, 40F05, 40G99.
Key words:
Absolute summability, Summability factors, Almost and power increasing sequences, Infinite series.
Abstract:
In the present paper, a general theorem on | N̄ , pn |k summability factors of
infinite series has been proved under more weaker conditions. Also we have
obtained a new result concerning the | C, 1 |k summability factors.
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Contents
1
Introduction
3
2
Known Results
5
3
Main Result
7
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
Title Page
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1.
Introduction
A positive sequence (bn ) is said to be almost increasing if there exists a positive
increasing sequence (cn ) and two positive constants A and B such that Acn ≤ bn ≤
Bcn (see [1]). We denote by BVO the expression BV ∩ CO , where CO and BV are
the set of all nullP
sequences and the set of all sequences with bounded variation,
respectively. Let
an be a given infinite series with partial sums (sn ). We denote
α
α
by un and tn the n-th Cesàro means of order α, with α > −1, of the sequences (sn )
and (nan ), respectively, i.e.,
uαn
(1.1)
tαn
(1.2)
n
1 X α−1
A sv ,
= α
An v=0 n−v
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where
Aαn = O(nα ), α > −1, Aα0 = 1 and Aα−n = 0 for
P
The series an is said to be summable |C, α|k , k ≥ 1, if (see [6, 8])
∞
X
(1.4)
n=1
n
k−1
n > 0.
∞
X
α
|tαn |k
un − uαn−1 k =
< ∞.
n
n=1
If we take α = 1, then we get |C, 1|k summability.
Let (pn ) be a sequence of positive numbers such that
(1.5)
Pn =
n
X
v=0
pv → ∞ as n → ∞,
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
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n
1 X α−1
= α
A vav ,
An v=1 n−v
(1.3)
Increasing Sequences
(P−i = p−i = 0, i ≥ 1).
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The sequence-to-sequence transformation
n
1 X
σn =
p v sv
Pn v=0
(1.6)
defines the sequence (σn ) of the Riesz mean or simply the (N̄ , pn ) mean of the
sequence (sn ), generated by the
of coefficients (pn ) (see [7]). The series
sequence
P
an is said to be summable N̄ , pn k , k ≥ 1, if (see [2, 3])
(1.7)
∞
X
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
(Pn /pn )k−1 |∆σn−1 |k < ∞,
n=1
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where
n
pn X
Pv−1 av , n ≥ 1.
(1.8)
∆σn−1 = −
Pn Pn−1 v=1
In the special case pn = 1 for all values of n, N̄ , pn k summability is the same as
|C, 1|k summability.
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2.
Known Results
Mishra and Srivastava [10] have proved the following theorem concerning the N̄ , pn summability factors.
Theorem A. Let (Xn ) be a positive non-decreasing sequence and let there be sequences (βn ) and (λn ) such that
|∆λn | ≤ βn ,
(2.1)
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
βn → 0 as
(2.2)
n → ∞,
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∞
X
(2.3)
n=1
|λn | Xn = O(1).
(2.4)
If
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n
X
(2.5)
v=1
|sv |
= O(Xn ) as
v
n→∞
and (pn ) is a sequence such that
(2.7)
P∞
n=1
n λn
an Pnp
n
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Pn = O(npn ),
(2.6)
then the series
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n |∆βn | Xn < ∞,
Pn ∆pn = O(pn pn+1 ),
is summable N̄ , pn .
Later on Bor [4] generalized Theorem A for N̄ , pn k summability in the following form.
Theorem B. Let (Xn ) be a positive non-decreasing sequence and the sequences
(βn ) and (λn ) are such that conditions (2.1) – (2.7) of Theorem A are satisfied with
the condition (2.5) replaced by:
n
X
|sv |k
(2.8)
v=1
v
= O(Xn ) as
n → ∞.
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
Then the series
P∞
Pn λn
N̄ , pn , k ≥ 1.
a
is
summable
n
n=1
npn
k
It may be noticed that if we take k = 1, then we get Theorem A.
Quite recently Bor [5] has proved Theorem B under weaker conditions by taking
an almost increasing sequence instead of a positive non-decreasing sequence.
Theorem C. Let (Xn ) be an almost increasing sequence.
(2.1)
P∞ If thePnconditions
λn
– (2.4) and (2.6) – (2.8) are satisfied, then the series n=1 an npn is summable
N̄ , pn , k ≥ 1.
k
Remark 1. It should be noted that, under the conditions of Theorem B, (λn ) is
bounded and ∆λn = O(1/n) (see [4]).
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3.
Main Result
The aim of this paper is to prove Theorem C under weaker conditions. For this
we need the concept of quasi β-power increasing sequences. A positive sequence
(γn ) is said to be a quasi β-power increasing sequence if there exists a constant
K = K(β, γ) ≥ 1 such that
Knβ γn ≥ mβ γm
(3.1)
Increasing Sequences
Hüseyin Bor
holds for all n ≥ m ≥ 1. It should be noted that almost every increasing sequence is
a quasi β-power increasing sequence for any nonnegative β, but the converse need
not be true as can be seen by taking the example, say γn = n−β for β > 0.
Now we shall prove the following theorem.
vol. 8, iss. 3, art. 82, 2007
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Theorem 3.1. Let (Xn ) be a quasi β-power increasing sequence for some 0 < β <
1. If the conditions (2.1) – (2.4), (2.6) – (2.8) and
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(λn ) ∈ BVO
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(3.2)
are satisfied, then the series
P∞
n=1
n λn
an Pnp
n
is summable N̄ , pn k , k ≥ 1.
It should be noted that if we take (Xn ) as an almost increasing sequence, then we
get Theorem C. In this case, condition (3.2) is not needed.
We require the following lemma for the proof of Theorem 3.1.
Lemma 3.2 ([9]). Except for the condition (3.2), under the conditions on (Xn ), (βn )
and (λn ) as taken in the statement of Theorem 3.1, the following conditions hold,
when (2.3) is satisfied:
(3.3)
nXn βn = O(1),
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∞
X
(3.4)
βn Xn < ∞.
n=1
P
Proof of Theorem 3.1. Let (Tn ) denote the (N̄ , pn ) mean of the series ∞
n=1
Then, by definition, we have
n
v
n
1 X
1 X X ar P r λ r
av P v λ v
(3.5)
Tn =
pv
=
(Pn − Pv−1 )
,
Pn v=1 r=1 rpr
Pn v=1
vpv
an Pn λn
.
npn
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
and thus
(3.6)
Tn − Tn−1
n
pn X Pv−1 Pv av λv
=
,
Pn Pn−1 v=1
vpv
n ≥ 1.
Using Abel’s transformation, we get
n
pn X
Pv−1 Pv λv
λn sn
Tn − Tn−1 =
sv ∆
+
Pn Pn−1 v=1
vpv
n
n−1
s n λn
pn X Pv+1 Pv ∆λv
=
+
sv
n
Pn Pn−1 v=1 (v + 1)pv+1
n−1
n−1
Pv
pn X
1
pn X
Pv s v λ v ∆
sv Pv λv
+
−
Pn Pn−1 v 1
vpv
Pn Pn−1 v=1
v
=
= Tn,1 + Tn,2 + Tn,3 + Tn,4 ,
say.
To prove 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.
(3.7)
pn
n=1
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Firstly by using Abel’s transformation, we have
k−1
k−1
m m X
X
Pn
Pn
|sn |k
k
|Tn,1 | =
|λn |k−1 |λn |
pn
npn
n
n=1
n=1
= O(1)
= O(1)
= O(1)
= O(1)
m
X
n=1
m−1
X
n=1
m−1
X
n=1
m−1
X
|λn |
| |sn | |k
n
∆ |λn |
n
X
|sv |k
v=1
v
+ O(1) |λm |
m
X
|sn |k
n=1
n
|∆λn | Xn + O(1) |λm | Xm
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
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βn Xn + O(1) |λm | Xm = O(1)
as m → ∞,
n=1
by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2.
Now, using the fact that Pv+1 = O((v + 1)pv+1 ), by (2.6), and then applying
Hölder’s inequality, we have
n−1
k
m+1
m+1
X Pn k−1
X
X
pn k
|Tn,2 | = O(1)
Pv sv ∆λv k
pn
P P
n=2
n=2 n n−1 v=1
(
)k
m+1
n−1
X pn
X
Pv
|sv | pv |∆λv |
= O(1)
k
p
P
P
v
n
n−1
v=1
n=2
!k−1
m+1
n−1
n−1
X p n X Pv k
1 X
k
k
|sv | pv |∆λv |
pv
= O(1)
P P
pv
Pn−1 v=1
n=2 n n−1 v=1
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= O(1)
= O(1)
k
m X
Pv
pv
v=1
m
X
Pv
v=1
= O(1)
= O(1)
= O(1)
= O(1)
= O(1)
m
X
v=1
m
X
v=1
m−1
X
v=1
m−1
X
v=1
m−1
X
|sv |k pv |∆λv |k
|∆λv |
pv
k−1
k
|sv | |∆λv |
m+1
X
pn
P P
n=v+1 n n−1
|sv |k |∆λv |
Pv
vpv
k−1
Increasing Sequences
Hüseyin Bor
k
vβv
|sv |
v
∆(vβv )
vol. 8, iss. 3, art. 82, 2007
v
X
|sr |k
r=1
r
+ O(1)mβm
m
X
|sv |k
v=1
v
|∆(vβv )| Xv + O(1)mβm Xm
v |∆βv | Xv + O(1)
v=1
m−1
X
|βv | Xv + O(1)mβm Xm
n=2
k−1
|Tn,3 |k = O(1)
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as m → ∞, in view of the hypotheses of Theorem 3.1 and Lemma 3.2.
Pv
Again, since ∆( vp
) = O( v1 ), by (2.6) and (2.7) (see [10]), as in Tn,1 we have
v
Pn
pn
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v=1
= O(1)
m+1
X
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m+1
X
n=2
pn
k
Pn Pn−1
( n−1
X
1
Pv |sv | |λv |
v
v=1
)k
Close
= O(1)
m+1
X
n=2
m+1
X
pn
k
Pn Pn−1
( n−1 X Pv
1
pv |sv | |λv |
v
)k
pv
(
)k−1
k
n−1 n−1
X
1 X
Pv
k
k
pv |sv | |λv |
pv
vpv
Pn−1 v=1
v=1
v=1
pn
P P
n=2 n n−1
k
m+1
m X
X
pn
Pv
k
k
|sv | pv |λv |
= O(1)
vpv
P P
n=v+1 n n−1
v=1
m
k
X
1 v
Pv
= O(1)
pv |sv |k |λv |k .
vpv
Pv v
v=1
m
k−1
X
Pv
|sv |k
= O(1)
|λv |k−1 |λv |
vpv
v
v=1
= O(1)
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
Title Page
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m
X
|sv |k
= O(1)
|λv |
v
v=1
= O(1)
m−1
X
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Xv βv + O(1)Xm |λm | = O(1)
v=1
Finally, using Hölder’s inequality, as in Tn,3 we have
k
m+1
m+1
n−1
X Pn k−1
X pn X
P
v
| Tn,4 |k =
s
λ
v
v
k
pn
v P P
n=2
n=2 n n−1 v=1
k
m+1
n−1
X pn X
Pv
=
sv
p v λv k
vpv
P P
n=2 n n−1 v=1
as
m → ∞.
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m+1
X
k
n−1
pn X
Pv
k
pv |λv |k
≤
|sv |
P
P
vp
v
n=2 n n−1 v=1
m
k
X Pv
1 v
= O(1)
|sv |k pv |λv |k
·
vpv
Pv v
v=1
= O(1)
= O(1)
m
X
v=1
m−1
X
|λv |
1
n−1
X
Pn−1
v=1
!k−1
pv
|sv |k
v
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
Xv βv + O(1)Xm |λm | = O(1)
as m → ∞.
v=1
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Therefore we get
k−1
m X
Pn
|Tn,r |k = O(1)
p
n
n=1
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as
m → ∞,
for
r = 1, 2, 3, 4.
This completes the proof of Theorem 3.1.
Finally if we take pn = 1 for all values of n in the theorem, then we obtain a new
result concerning the |C, 1|k summability factors.
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References
[1] S. ALJANCIC AND D. ARANDELOVIC, O-regularly varying functions, Publ.
Inst. Math., 22 (1977), 5–22.
[2] H. BOR, On two summability methods, Math. Proc. Camb. Philos Soc., 97
(1985), 147–149.
[3] H. BOR, A note on two summability methods, Proc. Amer. Math. Soc., 98
(1986), 81–84.
[4] H. BOR, A note on N̄ , pn k summability factors of infinite series, Indian J.
Pure Appl. Math., 18 (1987), 330–336.
Increasing Sequences
Hüseyin Bor
vol. 8, iss. 3, art. 82, 2007
Title Page
[5] H. BOR, A new application of almost increasing sequences, J. Comput. Anal.
Appl., (in press).
[6] T.M. FLETT, On an extension of absolute summability and some theorems of
Littlewood and Paley, Proc. London Math. Soc., 7 (1957), 113–141.
[7] G.H. HARDY, Divergent Series, Oxford Univ. Press., Oxford, (1949).
[8] E. KOGBETLIANTZ, Sur les séries absolument sommables par la méthode des
moyennes arithmétiques, Bull. Sci. Math., 49 (1925), 234–256.
[9] L. LEINDLER, A new application of quasi power increasing sequences, Publ.
Math. Debrecen, 58 (2001), 791–796.
[10] 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.
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