IJMMS 27:1 (2001) 7–15
PII. S0161171201010419
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON AN APPLICATION OF ALMOST INCREASING SEQUENCES
HÜSEYİN BOR
(Received 3 May 2000 and in revised form 25 October 2000)
Abstract. Using an almost increasing sequence, a result of Mazhar (1977) on |C, 1|k summability factors has been generalized for |C, α; β|k and |N̄, pn ; β|k summability factors
under weaker conditions.
2000 Mathematics Subject Classification. 40D15, 40F05, 40G05.
1. Introduction. A sequence of (bn ) of positive numbers is said to be δ-quasimonotone, if bn → 0, bn > 0 ultimately and ∆bn ≥ −δn , where (δn ) is a sequence
of positive numbers (see [2]). Let
an be a given infinite series with (sn ) as the
α
denote the nth (C, α) means of the
sequence of its nth partial sums. Let σnα and tn
sequences (sn ) and (nan ), respectively, that is,
n
1 α−1
An−v sv ,
Aα
n v=0
(1.1)
n
1 α−1
An−v vav ,
Aα
n v=1
(1.2)
σnα =
α
=
tn
where
The series
α
Aα
,
n=O n
α > −1,
Aα
0 = 1,
Aα
−n = 0,
for n > 0.
(1.3)
an is said to be summable |C, α|k , k ≥ 1 and α > −1, if (see [6])
∞
∞
1
α k
t α k < ∞,
nk−1 σnα − σn−1
=
n
n
n=1
n=1
(1.4)
and it is said to be summable |C, α; β|k , k ≥ 1, α > −1 and β ≥ 0, if (see [7])
∞
∞
α k
α k
< ∞.
nβk+k−1 σnα − σn−1
=
nβk−1 tn
n=1
(1.5)
n=1
Let (pn ) be a sequence of positive numbers such that
Pn =
n
pv → ∞
as n → ∞, P−i = p−i = 0, i ≥ 1.
(1.6)
v=0
The sequence-to-sequence transformation
Tn =
n
1 pv s v
Pn v=0
(1.7)
8
HÜSEYİN BOR
defines the sequence (Tn ) of the Riesz mean or simply the (N̄, pn ) mean of the
sequence (sn ), generated by the sequence of coefficients (pn ) (see [8]).
The series an is said to be summable |N̄, pn |k , k ≥ 1, if (see [3])
∞ Pn k−1 ∆Tn−1 k < ∞,
p
n
n=1
(1.8)
and it is said to be summable |N̄, pn ; β|k , k ≥ 1, and β ≥ 0, if (see [4])
∞ Pn βk+k−1 ∆Tn−1 k < ∞,
p
n
n=1
(1.9)
where
∆Tn−1 = −
n
pn
Pv−1 av ,
Pn Pn−1 v=1
n ≥ 1.
(1.10)
In the special case when β = 0 (resp., pn = 1 for all values of n), |N̄, pn ; β|k summability
is the same as |N̄, pn |k (resp., |C, 1; β|k ) summability.
Also it is known that |C, α; β|k and |N̄, pn ; β|k summabilities are, in general,
independent of each other.
Mazhar [9] has proved the following theorem for |C, 1|k summability factors of
infinite series.
Theorem 1.1 (see [9]). Let λn → 0 as n → ∞. Suppose that there exists a sequence
of numbers (Bn ) such that it is δ-quasi-monotone with nδn log n < ∞, Bn log n is
convergent and |∆λn | ≤ |Bn | for all n. If
m
1
tn k = O(log m) as m → ∞,
n
n=1
(1.11)
where (tn ) is the nth (C, 1) mean of the sequence (nan ), then the series an λn is
summable |C, 1|k , k ≥ 1.
Remark 1.2. It should be noted that the condition “ nBn log n is convergent”
is enough to prove Theorem 1.1 rather than the conditions “ nδn log n < ∞ and
Bn log n is convergent.”
2. The main result. In view of Remark 1.2, the aim of this paper is to generalize
Theorem 1.1 for |C, α; β|k and |N̄, pn ; β|k summabilities under weaker conditions. For
this we need the concept of almost increasing sequence. A positive sequence (dn ) is
said to be almost increasing if there exists a positive increasing sequence (cn ) and
two positive constants A and B such that Acn ≤ dn ≤ Bcn (see [1]). Obviously, every
increasing sequence is almost increasing but the converse need not be true as can be
n
seen from the example dn = ne(−1) . Since log n is increasing, we are weakening the
hypotheses of the theorem replacing the increasing sequence by an almost increasing
sequence.
ON AN APPLICATION OF ALMOST INCREASING SEQUENCES
9
Now, we prove the following theorems.
Theorem 2.1. Let (Xn ) be an almost increasing sequence and λn → 0 as n → ∞.
Suppose that there exists a sequence of numbers (Bn ) such that it is δ-quasi-monotone
with nBn Xn convergent and |∆λn | ≤ |Bn | for all n. If the sequence (uα
n ), defined by
(see [10])
 α

tn
,
α = 1,
=
(2.1)
uα
α
n

 max tv , 0 < α < 1,
1≤v≤n
satisfies the condition
m
k
nβk−1 uα
= O Xm
n
as m → ∞,
(2.2)
n=1
then the series
an λn is summable |C, α; β|k , k ≥ 1 and 0 ≤ β < α ≤ 1.
Theorem 2.2. Let (Xn ) be an almost increasing sequence and λn → 0 as n → ∞.
Suppose that there exists a sequence of numbers (Bn ) such that it is δ-quasi-monotone
with nBn Xn convergent and |∆λn | ≤ |Bn | for all n. If (pn ) is a sequence such that
∞
Pv βk 1
Pn βk−1 1
=O
,
pn
Pn−1
pv
Pv
n=v+1
m Pn βk−1 tn k = O Xm
p
n
n=1
m Pn βk 1 tn k = O Xm
p
n
n
n=1
m λn n=1
then the series
n
= O(1)
as m → ∞,
(2.3)
as m → ∞,
as m → ∞,
an λn is summable |N̄, pn ; β|k for k ≥ 1 and 0 ≤ β < 1/k.
We need the following lemmas for the proof of our theorems.
Lemma 2.3 (see [5]). If 0 < α ≤ 1 and 1 ≤ v ≤ n, then
m
v
α−1
α−1
An−p ap ≤ max
Am−p ap .
1≤m≤v p=0
(2.4)
p=0
Under the conditions of Theorem 2.2 we obtain the following result.
Lemma 2.4. The following equation holds:
λn Xn = O(1) as n → ∞.
(2.5)
10
HÜSEYİN BOR
Proof. Since λn → 0 as n → ∞, we have
∞
∞
∞
∞
∆λv ≤
λn Xn = Xn ∆λv ≤ Xn
Xv ∆λv ≤
Xv Bv < ∞.
v=n
v=n
v=0
(2.6)
v=0
Hence |λn |Xn = O(1) as n → ∞.
3. Proof of Theorem 2.1. Let (Tnα ) be the nth (C, α), with 0 < α ≤ 1, mean of the
sequence (nan λn ). Then, by (1.1), we have
Tnα =
n
1 α−1
An−v vav λv .
Aα
n v=1
(3.1)
Applying Abel’s transformation, we get
v
n−1
n
1 λn α−1
∆λv
Aα−1
A
vav ,
n−p pap + α
α
An v=1
An v=1 n−v
p=1
Tnα =
(3.2)
so that making use of Lemma 2.3, we have
v
n
n−1
λ α
n α−1
α−1
T ≤ 1
∆λv A
pa
A
va
+
p
v
n
n−p
n−v
α
α An v=1
A
n
p=1
v=1
≤
n−1
1 α α
A u ∆λv + λn uα
n
α
An v=1 v v
(3.3)
α
α
= Tn,1
+ Tn,2
.
Since
α
T + T α k ≤ 2k T α k + T α k ,
n,1
n,2
n,1
n,2
(3.4)
to complete the proof of Theorem 2.1, it is enough to show that
∞
α k
<∞
nβk−1 Tn,r
for r = 1, 2.
(3.5)
n=1
Now, when k > 1, applying Hölder’s inequality with indices k and k , where 1/k+1/k =
1, we get
m+1
α k
nβk−1 Tn,1
n=2
≤
m+1
n=2
n
βk−1
−k
Aα
n
n−
v=1
k
α
Aα
v uv Bv
ON AN APPLICATION OF ALMOST INCREASING SEQUENCES
≤
m+1
n
βk−1
−k
Aα
n
n−1
k−1
Bv n−1
n=2
k α k Aα
uv Bv v
v=1
= O(1)
m+1
nβk−αk−1
n−1
n=2
m
11
v=1
k Bv v αk uα
v
v=1
k m+1
uα Bv αk
1
1+αk−βk
n
v=1
n=v+1
∞
m
dx
k
Bv = O(1)
v αk uα
v
1+αk−βk
x
v
v=1
= O(1)
v
m
= O(1)
v
m
k k
Bv = O(1)
v βk uα
v Bv v βk−1 uα
v
v
v=1
= O(1)
m−1
v=1
v
m
k
k
Bm ∆ v Bv r βk−1 uα
+
O(1)m
v βk−1 uα
r
v
r =1
v=1
= O(1)
m−1
v=1
∆ v Bv Xv + O(1)mBm Xm
v=1
= O(1)
m−1
m−1
v Bv Xv + O(1)
(v + 1)Bv+1 Xv+1 + O(1)mBm Xm
v=1
v=1
= O(1)
as m → ∞,
(3.6)
by virtue of the hypotheses of Theorem 2.1.
Finally, since |λn | = O(1), by hypothesis
m
m
α k k−1 βk−1 α k
=
λn n
nβk−1 Tn,2
un
n=1
n=1
= O(1)
∞
m
βk−1 α k λn n
∆λv un
v=n
n=1
∞
v
k
∆λv nβk−1 uα
= O(1)
v
v=1
(3.7)
n=1
∞
Bv Xv < ∞,
= O(1)
v=1
by virtue of the hypotheses of Theorem 2.1.
Therefore, we get
m
α k
= O(1)
nβk−1 Tn,r
as m → ∞, for r = 1, 2.
(3.8)
n=1
This completes the proof of Theorem 2.1.
Remark 3.1. It is natural to ask whether our theorem is true with α > 1. All we
α
depends
can say with certainty is that our proof fails if α > 1, for our estimate of Tn,1
upon Lemma 2.3, and Lemma 2.3 is known to be false when α > 1 (see [5] for details).
12
HÜSEYİN BOR
Proof of Theorem 2.2 . 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
a n λn .
(3.9)
Then, for n ≥ 1, we have
Tn − Tn−1 =
n
n
pn
pn
Pv−1 λv
vav .
Pv−1 av λv =
Pn Pn−1 v=1
Pn Pn−1 v=1
v
(3.10)
By Abel’s transformation, we have
Tn − Tn−1 =
n−1
n+1
pn
v +1
pn tn λn −
pv t v λ v
nPn
Pn Pn−1 v=1
v
+
n−1
n−1
pn
v +1
1
pn
+
Pv ∆λv tv
Pv tv λv+1
Pn Pn−1 v=1
v
Pn Pn−1 v=1
v
(3.11)
= Tn,1 + Tn,2 + Tn,3 + Tn,4 .
Since
Tn,1 + Tn,2 + Tn,3 + Tn,4 k ≤ 4k Tn,1 k + Tn,2 k + Tn,3 k + Tn,4 k ,
(3.12)
to complete the proof of Theorem 2.2, it is enough to show that
∞ Pn βk+k−1 Tn,r k < ∞ for r = 1, 2, 3, 4.
p
n
n=1
(3.13)
Since (λn ) → 0 as n → ∞ by the hypothesis of Theorem 2.2, we have
m m Pn βk+k−1 Pn βk−1 Tn,1 k = O(1)
λn k−1 λn tn k
pn
pn
n=1
n=1
m
Pn βk−1 k
λn tn = O(1)
pn
n=1
n Pv βk−1 tv k
∆λn p
v
n=1
v=1
m Pn βk−1 tn k
+ O(1)λm pn
n=1
= O(1)
= O(1)
m−1
m−1
∆λn Xn + O(1)λm Xm
n=1
= O(1)
m−1
Bn Xn + O(1)λm Xm = O(1)
as m → ∞,
n=1
(3.14)
by virtue of the hypotheses of Theorem 2.2 and in view of Lemma 2.4.
ON AN APPLICATION OF ALMOST INCREASING SEQUENCES
13
Now, when k > 1, applying Hölder’s inequality with indices k and k , where 1/k +
1/k = 1, as in Tn,1 , we have
m+1
n=2
Pn
pn
βk+k−1
n−1
m+1
Pn βk−1 1
k k
Tn,2 k = O(1)
pv λv tv pn
Pn−1 v=1
n=2
k−1
n−1
1 pv
×
Pn−1 v=1
Pn βk−1 1
k−1 k m+1
= O(1)
pv λv λv tv pn
Pn−1
v=1
n=v+1
βk−1
m k Pv
tv λv = O(1) as m → ∞.
= O(1)
p
v
v=1
m
(3.15)
Again, we have
m+1
n=2
Pn
pn
βk+k−1
n−1
m+1
Pn βk−1 1
k
Tn,3 k = O(1)
Pv Bv tv
pn
Pn−1 v=1
n=2
k−1
n−1
1 Pv Bv
×
Pn−1 v=1
Pn βk−1 1
k m+1
Pv Bv tv pn
Pn−1
v=1
n=v+1
m
Pv βk k
Bv tv = O(1)
p
v
v=1
m
Pv βk 1 k
tv = O(1)
v Bv pv
v
v=1
= O(1)
= O(1)
m
m−1
v
∆ v Bv v=1
i=1
Pi
pi
βk
1
ti k
i
m Pv βk 1 tv k
+ O(1)mBm pv
v
v=1
= O(1)
m−1
∆ v Bv Xv + O(1)mBm Xm
v=1
= O(1)
m−1
m−1
vXv Bv + O(1)
(v + 1)Bv+1 Xv+1
v=1
+ O(1)mBm Xm
= O(1)
as m → ∞,
by virtue of the hypotheses of Theorem 2.2.
v=1
(3.16)
14
HÜSEYİN BOR
Finally, we have
m+1
n=2
Pn
pn
βk+k−1
m+1
λv+1 k
Pn βk−1 1 n−1
tv Tn,4 k = O(1)
Pv
pn
Pn−1 v=1
v
n=2
n−1
λv+1 k−1
1 ×
Pv
Pn−1 v=1
v
m
λv+1 k m+1
Pn βk−1 1
tv = O(1)
Pv
pn
Pn−1
v
v=1
n=v+1
m
βk
1
tv k
λv+1 Pv
= O(1)
p
v
v
v=1
= O(1)
m−1
v
∆λv+1 v=1
r =1
m Pv
+ O(1)λm+1 p
v
v=1
= O(1)
m−1
Pr
pr
βk
βk
1
tr k
r
(3.17)
1
tv k
v
∆λv+1 Xv+1 + O(1)λm+1 Xm+1
v=1
= O(1)
m−1
Bv+1 Xv+1 + O(1)λm+1 Xm+1
v=1
= O(1)
as m → ∞,
by virtue of the hypotheses of Theorem 2.2 and in view of Lemma 2.4.
Therefore, we get
m Pn βk+k−1 Tn,r k = O(1) as m → ∞, for r = 1, 2, 3, 4.
p
n
n=1
(3.18)
This completes the proof of Theorem 2.2.
If we take pn = 1 for all values of n in this theorem, then we get a result concerning
the |C, 1; β|k summability factors.
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Hüseyi̇n Bor: Department of Mathematics, Erciyes University 38039, Kayseri, Turkey
E-mail address: bor@erciyes.edu.tr