ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII “AL.I. CUZA” DIN IAŞI (S.N.)
MATEMATICĂ, Tomul LXI, 2015, f.1
DOI: 10.2478/aicu-2013-0049
A NEW APPLICATION OF ALMOST INCREASING
SEQUENCES
BY
H.S. ÖZARSLAN and A. KETEN
Abstract. Bor has proved a main theorem dealing with |N̄ , pn |k summability factors
of infinite series. In this paper, we have generalized this theorem to the ϕ − |A, pn |k
summability factors, under weaker conditions by using an almost increasing sequence
instead of a positive monotonic non-decreasing sequence.
Mathematics Subject Classification 2010: 40D25, 40F05, 40G99.
Key words: absolute matrix summability, almost increasing sequences, infinite series.
1. Introduction
A positive sequence (bn ) is said to be almost increasing if there exists a
positive increasing sequence (cn ) and
Ptwo positive constants A and B such
that Acn ≤ bn ≤ Bcn (see [1]). Let
an be a given infinite series with the
partial sums (sn ). Let (pn ) be a sequence of positive numbers such that
(1)
Pn =
n
X
pv → ∞
as
n → ∞,
(P−i = p−i = 0,
i ≥ 1) .
v=0
The sequence-to-sequence transformation
(2)
n
1 X
σn =
pv s v
Pn
v=0
defines the sequence (σn ) of the N̄ , pn mean of the sequenceP(sn ), generated by the sequence of coefficients (pn ) (see [5]). The series
an is said
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H.S. ÖZARSLAN and A. KETEN
to be summable N̄ , pn k , k ≥ 1, if (see [2])
∞ X
Pn k−1
(3)
n=1
pn
|σn − σn−1 |k < ∞.
Let A = (anv ) be a normal matrix, i.e., a lower triangular matrix of
nonzero diagonal entries. Then A defines the sequence-to-sequence transformation, mapping the sequence s = (sn ) to As = (An (s)), where
(4)
An (s) =
n
X
anv sv ,
n = 0, 1, ...
v=0
The series
P
an is said to be summable |A, pn |k , k ≥ 1, if (see [7])
(5)
∞ X
Pn k−1
n=1
pn
¯ n (s)|k < ∞,
|∆A
¯ n (s) = An (s) − An−1 (s). Let (ϕn ) be any sequence of positive real
where ∆A
P
numbers. We say that the series
an is summable ϕ − |A, pn |k , k ≥ 1, if
∞
X
(6)
n=1
¯ n (s)k < ∞.
ϕnk−1 ∆A
If we take ϕn = Ppnn , then ϕ − |A, pn |k summability reduces to |A, pn |k
summability. Also, if we take ϕn = Ppnn and anv = Ppvn , then we get N̄ , pn k
summability. Furthermore, if we take ϕn = n and anv = Ppvn and pn = 1 for
all values of n, then ϕ − |A, pn |k summability reduces to |C, 1|k summability
(see [4]).
Before stating the main theorem we must first introduce some further
notations.
Given a normal matrix A = (anv ), we associate two lower semi matrices
Ā = (ānv ) and  = (ânv ) as follows:
(7)
ānv =
n
X
ani ,
n, v = 0, 1, ...
i=v
(8)
â00 = ā00 = a00 ,
ânv = ānv − ān−1,v ,
n = 1, 2, ...
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A NEW APPLICATION OF ALMOST INCREASING SEQUENCES
155
It may be noted that Ā and  are the well-known matrices of series-tosequence and series-to-series transformations, respectively. Then, we have
(9)
An (s) =
¯ n (s) =
∆A
(10)
n
X
v=0
n
X
anv sv =
n
X
ānv av ,
v=0
ânv av .
v=0
2. Known result
Bor [3] has proved the following theorem for N̄ , pn k summability method.
Theorem A. Let (pn ) be a sequence of positive numbers such that
(11)
Pn = O(npn )
n → ∞.
as
If (Xn ) is a positive monotonic non-decreasing sequence such that
(12)
λm Xm = O(1) as m → ∞,
m
X
nXn |∆2 λn | = O(1),
(13)
n=1
m
X
(14)
n=1
1
where tn = n+1
Pn
v=1
pn
|tn |k = O(Xm )
Pn
vav then the series
P
as
m → ∞,
an λn is summable |N̄ , pn |k , k≥1.
3. The main result
The aim of this paper is to generalize Theorem A to ϕ − |A, pn |k summability. Now we shall prove the following theorem.
Theorem. Let A = (anv ) be a positive normal matrix such that
(15)
ān0 = 1,
(16)
an−1,v ≥ anv , f or n ≥ v + 1,
pn
ann = O
,
Pn
|ân,v+1 | = O (v |∆v ânv |) .
(17)
(18)
n = 0, 1, ...,
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H.S. ÖZARSLAN and A. KETEN
Let (Xn ) be an almost increasing sequence and ( ϕPnnpn ) be a non-increasing
sequence. If conditions (11)-(13) of Theorem A and
k
m
X
pn
k−1
ϕn
(19)
|tn |k = O(Xm ) as m → ∞,
P
n
n=1
P
are satisfied, then the series
an λn is summable ϕ − |A, pn |k , k ≥ 1.
It should be noted that if we take (Xn ) as a positive monotonic nondecreasing sequence, anv = Ppvn and ϕn = Ppnn in this theorem, then we get
Theorem A. In this case, condition (19) reduces to condition (14) and the
condition ”( ϕPnnpn ) is a non-increasing sequence” is automatically satisfied.
We require the following lemma for the proof of the theorem.
Lemma ([6]). Under the conditions on (Xn ) and (λn ) which are taken
in the statement of our theorem, then we have the following:
(20)
nXn |∆λn | = O(1) as n → ∞,
∞
X
Xn |∆λn | < ∞.
(21)
n=1
4. Proof of the theorem
P
Let (Tn ) denotes A-transform of the series
an λn . Then, by (9) and
(10), we have
¯ n=
∆T
n
X
ânv av λv =
n
X
ânv λv
v=1
v=1
v
vav .
Applying Abel’s transformation to this sum, we get
¯ n
∆T
=
n−1
X
∆v
v=1
=
ânv λv
v
X
v
rar +
r=1
n−1
X
n
ânn λn X
rar
n
r=1
v+1
n+1
ann λn tn +
∆v (ânv ) λv tv
n
v
v=1
+
n−1
X
v=1
=
n−1
X1
v+1
ân,v+1 ∆λv tv +
ân,v+1 λv+1 tv
v
v
v=1
Tn,1 + Tn,2 + Tn,3 + Tn,4 ,
say.
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A NEW APPLICATION OF ALMOST INCREASING SEQUENCES
157
To prove
theorem, by Minkowski’s inequality, it is sufficient to show
P thek−1
that ∞
ϕ
|Tn,r |k < ∞, for r = 1, 2, 3, 4. Firstly, we have that
n=1 n
k
m
m
X
X
k−1 n + 1
k−1
k
ann λn tn ϕn ϕn |Tn,1 | =
n
n=1
n=1
k
k
m
m
X
X
pn
pn
k−1
k
k−1
k−1
k
ϕn |λn ||tn |
= O(1)
ϕn |λn | |λn ||tn |
= O(1)
Pn
Pn
n=1
n=1
k
k
n
m−1
m
X
X
X
pr
pv
k−1
k−1
k
ϕr
∆|λn |
= O(1)
ϕv
|tr | + O(1)|λm |
|tv |k
Pr
Pv
= O(1)
= O(1)
n=1
m−1
X
n=1
m−1
X
r=1
v=1
∆|λn |Xn + O(1)|λm |Xm
|∆λn |Xn + O(1)|λm |Xm = O(1)
as
m → ∞,
n=1
by virtue of the hypotheses of the theorem and lemma.
Applying Hölder’s inequality with indices k and k′ , where k > 1 and
1
1
k + k ′ = 1, as in Tn,1 , we have that
k
n−1
m+1
m+1
X v + 1
X
X
k−1
k−1
k
ϕn ∆v (ânv )λv tv ϕn |Tn,2 | =
v
v=1
n=2
n=2
!k−1
!
n−1
n−1
m+1
X
X
X
|∆v (ânv )|
|∆v (ânv )||λv |k |tv |k ×
ϕnk−1
= O(1)
= O(1)
= O(1)
n=2
m+1
X
n=2
m
X
ϕn pn
Pn
k−1
|λv |
= O(1)
k
|λv ||tv |
= O(1)
k
|λv | |tv |
as
k
|∆v (ânv )||λv | |tv |
v=1
m+1
X
ϕv pv
Pv
|λv | |tv |k ϕvk−1
v=1
= O(1)
k
n=v+1
v=1
m
X
n−1
X
k−1
v=1
m
X
v=1
v=1
ϕn pn
Pn
k−1 m+1
X
k−1
!
|∆v (ânv )|
|∆v (ânv )|
n=v+1
pv
Pv
k
m → ∞,
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H.S. ÖZARSLAN and A. KETEN
by virtue of the hypothesesof the
theorem and lemma.
1
Now, since v|∆λv | = O Xv = O(1), by (20), we have that
n−1
k
X v + 1
ϕnk−1 ϕnk−1 |Tn,3 | =
ân,v+1 ∆λv tv v
v=1
n=2
n=2
!k−1
!
n−1
n−1
m+1
X
X
X
|ân,v+1 ||∆λv |
|ân,v+1 ||∆λv ||tv |k ×
ϕnk−1
= O(1)
m+1
X
= O(1)
= O(1)
m+1
X
k
n=2
v=1
m+1
X
n−1
X
ϕnk−1
n=2
m+1
X
n=2
v=1
v|∆v ânv ||∆λv ||tv |k
v=1
ϕn pn
Pn
k−1
m
X
n−1
X
!
n−1
X
×
v|∆v ânv ||∆λv |
v=1
k
v|∆v ânv ||∆λv ||tv |
v=1
m+1
X !k−1
!
ϕn pn k−1
v|∆λv ||tv |
= O(1)
|∆v ânv |
Pn
n=v+1
v=1
m+1
m
X
ϕv pv k−1 X
|∆v ânv |
v|∆λv ||tv |k
= O(1)
P
v
n=v+1
v=1
k
m
X
pv
v|∆λv ||tv |k ϕvk−1
= O(1)
Pv
v=1
k
v
m−1
X
X
pr
k−1
ϕr
|∆ (v|∆λv |)|
= O(1)
|tr |k
Pr
r=1
v=1
k
m
X
pv
k−1
|tv |k
ϕv
+ O(1)m|∆λm |
Pv
k
v=1
= O(1)
m−1
X
2
vXv |∆ λv | + O(1)
as
Xv |∆λv+1 | + O(1)m|∆λm |Xm
v=1
v=1
= O(1)
m−1
X
m → ∞,
by virtue of the hypotheses of the theorem and lemma.
Finally, using the fact Pn = O(npn ), by (11), as in Tn,1 , we have that
k
m+1
m+1
n−1
X
X
X
1
ϕnk−1 ϕnk−1 |Tn,4 |k =
ân,v+1 λv+1 tv v
n=2
n=2
v=1
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A NEW APPLICATION OF ALMOST INCREASING SEQUENCES
= O(1)
m+1
X
ϕnk−1
v=1
n=2
= O(1)
m+1
X
n−1
X
ϕnk−1
n=2
m+1
X
n−1
X
v=1
1
|ân,v+1 ||λv+1 |k |tv |k
v
!
×
1
|ân,v+1 ||λv+1 |k |tv |k
v
!
×
159
!k−1
1
|ân,v+1 |
v
v=1
!k−1
n−1
X
|∆v ânv |
n−1
X
v=1
!
n−1
ϕn pn k−1 X 1
= O(1)
|ân,v+1 ||λv+1 |k |tv |k
Pn
v
n=2
v=1
m
m+1
X1
X ϕn pn k−1
k−1
k
= O(1)
|ân,v+1 |
|λv+1 | |λv+1 ||tv |
v
Pn
v=1
n=v+1
k−1 m+1
m
X
X
1
k ϕv pv
|ân,v+1 |
|λv+1 ||tv |
= O(1)
v
Pv
n=v+1
v=1
m
X
1
ϕv pv k−1
= O(1)
|λv+1 ||tv |k
v
Pv
v=1
k
m
X
pv
Pv
k k−1
|λv+1 ||tv | ϕv
= O(1)
vp
P
v
v
v=1
m
k
X
pv
|λv+1 ||tv |k ϕvk−1
= O(1)
P
v
v=1
= O(1)
as
m → ∞,
by virtue of the hypotheses of the theorem and lemma. This completes the
proof of the theorem.
Corollary 1. If we take ϕn =
|A, pn |k summability.
Corollary 2. If we take anv =
with |N̄ , pn , ϕn |k summability.
Pn
pn ,
pv
Pn ,
then we get a result concerning the
then we have another a result dealing
Corollary 3. If we take anv = Ppvn and pn = 1 for all values of n, then
we get a result dealing with |C, 1, ϕn |k summability.
Corollary 4. If we take ϕn = n, anv = Ppvn and pn = 1 for all values of
n, then we get a result for |C, 1|k summability.
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H.S. ÖZARSLAN and A. KETEN
Corollary 5. If we take k = 1 and anv = Ppvn , then we get a result
for |N̄ , pn | summability and in this case the condition ”( ϕPnnpn ) is a nonincreasing sequence” is not needed.
REFERENCES
1. Bari, N.K.; Stečkin, S.B. – Best approximations and differential properties of two
conjugate functions, (Russian) Trudy Moskov. Mat. Obšč., 5 (1956), 483–522.
2. Bor, H. – On two summability methods, Math. Proc. Cambridge Philos. Soc., 97
(1985), 147–149.
3. Bor, H. – On absolute summability factors, Proc. Amer. Math. Soc., 118 (1993),
71–75.
4. Flett, T.M. – On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc., 7 (1957), 113–141.
5. Hardy, G.H. – Divergent Series, Oxford, at the Clarendon Press, 1949.
6. Mazhar, S.M. – Absolute summability factors of infinite series, Kyungpook Math.
J., 39 (1999), 67–73.
7. Sulaiman, W.T. – Inclusion theorems for absolute matrix summability methods of
an infinite series. IV, Indian J. Pure Appl. Math., 34 (2003), 1547–1557.
Received: 27.III.2012
Revised: 6.VII.2012
Accepted: 13.VII.2012
Department of Mathematics,
Erciyes University,
38039, Kayseri,
TURKEY
[email protected]
[email protected]
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