IJMMS 25:5 (2001) 293–298
PII. S0161171201005105
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON ALMOST INCREASING SEQUENCES
AND ITS APPLICATIONS
HIKMET S. ÖZARSLAN
(Received 7 July 1999 and in revised form 18 April 2000)
Abstract. We prove a general theorem on |N̄, pn ; δ|k summability factors, which generalizes a theorem of Bor (1994) on |N̄, pn |k summability factors, under weaker conditions
by using an almost increasing sequence.
2000 Mathematics Subject Classification. Primary 40D15, 40F05, 40G99.
1. Introduction. Let an be a given infinite series with partial sums (sn ). Let (pn )
be a sequence of positive numbers such that
Pn =
n
pv → ∞
as n → ∞,
P−i = p−i = 0, i ≥ 1 .
(1.1)
v=0
The sequence-to-sequence transformation
Tn =
n
1 pv s v
Pn v=0
(1.2)
defines the sequence (Tn ) of the (N̄, pn ) mean of the sequence (sn ), generated by the
sequence of coefficients (pn ) (see [4]). The series an is said to be summable |N̄, pn |k ,
k ≥ 1, if (see [1])
∞ Pn k−1
|∆Tn−1 |k < ∞
(1.3)
pn
n=1
and it is said to be summable |N̄, pn ; δ|k , k ≥ 1 and δ ≥ 0, if (see [2])
∞ Pn δk+k−1
|∆Tn−1 |k < ∞,
pn
n=1
(1.4)
where
∆Tn−1 = −
n
pn
Pv−1 av ,
Pn Pn−1 v=1
n ≥ 1.
(1.5)
In the special case when δ = 0, (respectively, pn = 1 for all values of n) |N̄, pn ; δ|k
summability is the same as |N̄, pn |k (respectively, |C, 1; δ|k ) summability.
294
HIKMET S. ÖZARSLAN
Mishra and Srivastava [6] proved the following theorem for |C, 1|k summability.
Theorem 1.1. Let (Xn ) be a positive nondecreasing sequence and let there be sequences (βn ) and (λn ) such that
∆λn ≤ βn ,
βn → 0 as n → ∞,
λn Xn = O(1) as n → ∞,
∞
n∆βn Xn < ∞.
(1.6)
(1.7)
(1.8)
(1.9)
n=1
If
m
1
sn k = O Xm
n
n=1
then the series
as m → ∞,
(1.10)
an λn is summable |C, 1|k , k ≥ 1.
Bor [3] has generalized Theorem 1.1 for |N̄, pn |k summability in the form of the
following theorem.
Theorem 1.2. Let (Xn ) be a positive nondecreasing sequence and the sequences (βn )
and (λn ) such that conditions (1.6), (1.7), (1.8), and (1.9) of Theorem 1.1 are satisfied.
Furthermore, if (pn ) is a sequence of positive numbers such that
Pn = O npn
as n → ∞,
m
pn
|sn |k = O Xm
P
n
n=1
then the series
as m → ∞,
(1.11)
(1.12)
an λn is summable |N̄, pn |k , k ≥ 1.
It should be noted that, if we take pn = 1 for all values of n, then condition (1.12)
will be reduced to condition (1.10). Also, it can be noticed that in this case condition
(1.11) is obvious.
2. The main result. The aim of this paper is to generalize Theorem 1.2 for |N̄, pn ; δ|k
summability under weaker conditions. For this we need the concept of almost increasing sequence. 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 . Obviously every increasing sequence is almost increasing sequence
n
but the converse need not be true as can be seen from the example bn = ne(−1) . So
we are weakening the hypotheses of Theorem 1.2 replacing the increasing sequence
by an almost increasing sequence.
Now, we will prove the following theorem.
Theorem 2.1. Let (Xn ) be an almost increasing sequence and the sequences (βn )
and (λn ) such that conditions (1.6), (1.7), (1.8), and (1.9) of Theorem 1.1 are satisfied.
295
ON ALMOST INCREASING SEQUENCES AND ITS APPLICATIONS
If (pn ) is a sequence such that condition (1.11) of Theorem 1.2 is satisfied and
m Pn δk−1 sn k = O Xm
pn
n=1
as m → ∞,
(2.1)
∞
Pv δk 1
Pn δk−1 1
=O
,
pn
Pn−1
pv
Pv
n=v+1
then the series
(2.2)
an λn is summable |N̄, pn ; δ|k for k ≥ 1 and 0 ≤ δ < 1/k.
Remark 2.2. It may be noted that if we take (Xn ) as a positive nondecreasing
sequence and δ = 0 in this theorem, then we get Theorem 1.2. In this case, condition
(2.1) reduces to condition (1.12) and condition (2.2) reduces to
1
pn
=O
,
P
P
P
n
n−1
v
n=v+1
∞
(2.3)
which always holds.
We need the following lemma for the proof of our theorem.
Lemma 2.3 (see [5]). Under the conditions on (Xn ), (βn ), and (λn ) as taken in the
statement of the theorem, the following conditions hold, when (1.9) is satisfied:
nβn Xn = O(1)
∞
as n → ∞,
(2.4)
βn Xn < ∞.
(2.5)
n=1
Proof of Theorem 2.1. Let (Tn ) denotes the (N̄, pn ) mean of the series
Then, by definition and changing the order of summation, we have
Tn =
n
n
v
1 1 pv
ai λi =
Pn − Pv−1 av λv .
Pn v=0
P
n
v=0
i=0
an λn .
(2.6)
Then, for n ≥ 1, we have
Tn − Tn−1 =
n
pn
Pv−1 av λv .
Pn Pn−1 v=1
(2.7)
By Abel’s transformation, we have
Tn − Tn−1 =
n−1
pn
pn
∆ Pv−1 λv sv +
s n λn
Pn Pn−1 v=1
Pn
=−
n−1
n−1
pn
pn
pn
pv s v λ v +
Pv sv ∆λv +
sn λn
Pn Pn−1 v=1
Pn Pn−1 v=1
Pn
= Tn,1 + Tn,2 + Tn,3 ,
(2.8)
296
HIKMET S. ÖZARSLAN
where Tn,i , i = 1, 2, 3, denotes the ith term in the sum. Since
Tn,1 + Tn,2 + Tn,3 k ≤ 3k Tn,1 k + Tn,2 k + Tn,3 k ,
(2.9)
to complete the proof of the theorem, it is enough to show that
∞ Pn δk+k−1 Tn,r k < ∞ for r = 1, 2, 3.
p
n
n=1
(2.10)
Now, when k > 1 applying Hölder’s inequality with indices k and k , where 1/k+1/k =
1, we have
m+1
n=2
Pn
pn
δk+k−1

k
m+1
Pn δk−1 −k n−1
Tn,1 k ≤

pv sv λv 
Pn−1
pn
n=2
v=1
≤
m+1
n=2
Pn
pn
δk−1
1
Pn−1

k−1
n−1
k k
1
pv sv λv 
pv 
Pn−1 v=1
v=1
n−1
= O(1)
Pn δk−1 1
k k m+1
pv sv λv pn
Pn−1
v=1
n=v+1
= O(1)
m Pv δk−1 sv k λv λv k−1
pv
v=1
= O(1)
m Pv δk−1 sv k λv p
v
v=1
= O(1)
m
m−1
v
∆λv v=1
r =1
Pr
pr
δk−1
k
sr m Pv δk−1 sv k
+ O(1)λm p
v
v=1
= O(1)
m−1
∆λv Xv + O(1)λm Xm
v=1
= O(1)
m−1
βv Xv + O(1)λm Xm
v=1
= O(1)
as m → ∞,
(2.11)
by virtue of the hypotheses of Theorem 1.2 and Lemma 2.3.
Since vβv = O(1/Xv ) by (2.4), using the fact that Pv = O(vpv ) by (1.11), and
|∆λn | ≤ βn by (1.6), and after applying Hölder’s inequality again, we have
ON ALMOST INCREASING SEQUENCES AND ITS APPLICATIONS
m+1
n=2
Pn
pn
δk+k−1
297
k

m+1
Pn δk−1 −k n−1
Tn,2 k ≤

Pv ∆λv sv 
Pn−1
p
n
n=2
v=1
k

m+1
Pn δk−1 −k n−1

= O(1)
vpv βv sv 
Pn−1
pn
n=2
v=1

k−1
n−1
m+1
Pn δk−1 1 n−1
k k
1
= O(1)
pv 
vβv pv sv 
pn
Pn−1 v=1
Pn−1 v=1
n=2
= O(1)
m
vβv
v=1
= O(1)
m
vβv
v=1
= O(1)
m
vβv
k
Pn δk−1 1
k m+1
pv sv pn
Pn−1
n=v+1
k Pv δk−1 k
sv pv
k−1 vβv
v=1
= O(1)
m
vβv
v=1
= O(1)
Pv δk−1 k
sv pv
Pv δk−1 k
sv pv
m−1
v
∆ vβv
v=1
r =1
Pr
pr
δk−1
|sr |k
m Pv δk−1 sv k
+ O(1)mβm
pv
v=1
= O(1)
m−1
∆ vβv Xv + O(1)mβm Xm
v=1
= O(1)
m−1
m−1
v ∆βv Xv + O(1)
βv+1 Xv+1 + O(1)mβm Xm
v=1
= O(1)
v=1
as m → ∞,
(2.12)
by virtue of the hypotheses of Theorem 1.2 and Lemma 2.3.
Finally, using the fact that Pv = O(vpv ) by (1.11), as in Tn,1 , we have
m m Pn δk+k−1 Pn δk−1 Tn,3 k = O(1)
sn k λn pn
pn
n=1
n=1
(2.13)
= O(1) as m → ∞.
Therefore, we get (2.10) and this completes the proof of the theorem.
If we take pn = 1 for all values of n in this theorem, then we get a result concerning
the |C, 1; δ|k summability factors.
298
HIKMET S. ÖZARSLAN
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[1]
[2]
[3]
[4]
[5]
[6]
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Hikmet S. Özarslan: Department of Mathematics, Erciyes University 38039, Kayseri,
Turkey
E-mail address: seyhan@erciyes.edu.tr