IJMMS 2004:44, 2371–2376
PII. S0161171204303510
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
QUASI β-POWER INCREASING SEQUENCES
H. BOR and L. DEBNATH
Received 1 March 2003 and in revised form 20 October 2003
We prove a theorem of Mazhar (1999) on |N̄, pn |k summability factors under weaker conditions by using a quasi β-power increasing sequence instead of an almost increasing sequence.
2000 Mathematics Subject Classification: 40D15, 40F05, 40G99.
1. Introduction. A positive sequence (bn ) is said to be almost increasing if there
exist a positive increasing sequence (cn ) and two positive constants A and B such that
Acn ≤ bn ≤ Bcn (see [1]). Obviously, every increasing sequence is almost increasing.
However, the converse need not be true as can be seen by taking the example, say
n
bn = ne(−1) . Let an be a given infinite series with partial sums (sn ). Let (tn ) denote
the nth (C, 1) mean of the sequence (nan ). A series an is said to be summable |C, 1|k ,
k ≥ 1, if (see [6, 8])
∞ tn k
< ∞.
n
n=1
(1.1)
Let (pn ) be a sequence of positive numbers such that
Pn =
n
pv → ∞
as n → ∞, P−i = p−i = 0, i ≥ 1.
(1.2)
v=0
The sequence-to-sequence transformation
σn =
n
1 pv s v
Pn v=0
(1.3)
defines the sequence (σn ) of the (N̄, pn ) mean of the sequence (sn ), generated by the
sequence of coefficients (pn ) (see [7]). The series an is said to be summable |N̄, pn |k ,
k ≥ 1, if (see [3])
∞ Pn k−1 ∆σn−1 k < ∞,
p
n
n=1
(1.4)
where
∆σn−1 = −
n
pn
Pv−1 av ,
Pn Pn−1 v=1
n ≥ 1.
(1.5)
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H. BOR AND L. DEBNATH
In the special case when pn = 1 for all values of n, |N̄, pn |k summability is the same as
|C, 1|k summability. Also if we take pn = 1/(n + 1), then |N̄, pn |k summability reduces
to |N̄, 1/(n + 1)|k summability.
Mazhar [9] has proved the following theorem on |C, 1|k summability factors of an
infinite series.
Theorem 1.1. If (Xn ) is a positive nondecreasing sequence such that
λm Xm = O(1) as m → ∞,
m
nXn ∆2 λn = O(1) as m → ∞,
(1.6)
(1.7)
n=1
m tn k
= O Xm
as m → ∞,
n
n=1
then the series
(1.8)
an λn is summable |C, 1|k , k ≥ 1.
Bor [5] has extended Theorem 1.1 for |N̄, pn |k summability method in the following
form.
Theorem 1.2. Under the conditions (1.6), (1.7),
Pn = O npn ,
m
pn t n k = O Xm
P
n
n=1
the series
as m → ∞,
(1.9)
(1.10)
an λn is summable |N̄, pn |k , k ≥ 1.
For pn = 1, (1.10) is the same as (1.8), and (1.9) holds. In this case, Theorem 1.2
reduces to Theorem 1.1. Also if we assume that (npn ) = O(Pn ), then (1.10) is equivalent
to (1.8) and |N̄, pn |k is equivalent to the |C, 1|k summability (see [2, 4]). Hence, under
the additional assumption (npn ) = O(Pn ), Theorem 1.1 is equivalent to Theorem 1.2.
Quite recently, Mazhar [10] obtained a further generalization of Theorem 1.2 under
weaker conditions by using an almost increasing sequence instead of positive nondecreasing sequence. Also it is clear that (1.9) and (1.10) imply (1.8). On the other hand,
(1.9) implies that
m
Pn
= O Pm
as m → ∞.
n
n=1
(1.11)
It may be remarked that (1.9) implies (1.11), but the converse need not be true. His
theorem is as follows.
Theorem 1.3. If (Xn ) is an almost increasing sequence and the conditions (1.6), (1.7),
(1.8), (1.10), and (1.11) hold, then the series an λn is summable |N̄, pn |k , k ≥ 1.
2. The main result. The aim of this note is to prove Theorem 1.3 under weaker conditions. For this we need the concept of quasi β-power increasing sequence. A positive
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QUASI β-POWER INCREASING SEQUENCES
sequence (γn ) is said to be quasi β-power increasing sequence if there exists a constant
K = K(β, γ) ≥ 1 such that
Knβ γn ≥ mβ γm
(2.1)
holds for all n ≥ m ≥ 1. It should be noted that every almost 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. So we are weakening
the hypotheses of Theorem 1.3, replacing an almost increasing sequence by a quasi
β-power increasing sequence. Now, we will prove the following theorem.
Theorem 2.1. Let (Xn ) be a quasi β-power increasing sequence for some 0 < β < 1.
If the conditions (1.6), (1.7), (1.8), (1.10), and (1.11) are satisfied, then the series an λn
is summable |N̄, pn |k , k ≥ 1.
We need the following lemma for the proof of Theorem 2.1.
Lemma 2.2. If (Xn ) is a quasi β-power increasing sequence for some 0 < β < 1, then
under the conditions (1.6) and (1.7),
nXn ∆λn = O(1),
∞
(2.2)
Xn ∆λn < ∞.
(2.3)
n=1
Proof. The condition (1.6) implies that λn = O(1) and it is easy to see that (1.7)
implies that n∆λn = O(1). Thus ∆λn → 0, n → ∞. Since 0 < β < 1, for any v ≥ n we have
nXn ≤ KvXv , by (2.1). Hence, by (1.7), we get that
∞
∞
2 ∆ λv ≤ K
vXv ∆2 λv < ∞,
nXn ∆λn ≤ nXn
v=n
(2.4)
v=n
thus nXn |∆λn | = O(1) as n → ∞. Also,
∞
∞
∞
∞
v
2 2
∆ λ v Xn ∆λn =
Xn ∆ λv ≤
Xn
n=1
n=1
=
∞
v=1
≤K
v=n
v=1
n=1
v
v
∞
2 2 ∆ λ v ∆ λv Kv β Xv
nβ Xn n−β ≤
n−β
n=1
v=1
(2.5)
n=1
v
∞
∞
2 β
2 dx
∆ λ v v Xv
∆ λv K(β)vXv < ∞,
≤
K
β
x
1
v=1
v=1
where K(β) is a constant depending only on β. This completes the proof of the lemma.
3. Proof of Theorem 2.1. Let (Tn ) denote the (N̄, pn ) mean of the series
Then, by definition, and changing the order of summation, we have
Tn =
v
n
n
1 1 pv
ai λi =
Pn − Pv−1 av λv .
Pn v=0
Pn v=0
i=0
an λn .
(3.1)
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H. BOR AND L. DEBNATH
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
(3.2)
By Abel’s transformation, we have
Tn − Tn−1 =
n−1
n+1
pn
v +1
pn t n λ n −
pv t v λ v
nPn
Pn Pn−1 v=1
v
+
n−1
n−1
pn
pn
v +1
1
+
Pv ∆λv tv
Pv tv λv+1
Pn Pn−1 v=1
v
Pn Pn−1 v=1
v
(3.3)
= 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 ,
(3.4)
to complete the proof of Theorem 2.1, it is enough to show that
∞ Pn k−1 Tn,r k < ∞,
p
n
n=1
for r = 1, 2, 3, 4.
(3.5)
In view of (1.6), (λn ) is bounded. Hence, we have that
m+1
n=2
Pn
pn
k−1
m
m
pn k
pn Tn,1 k =
λn k−1 λn tn k = O(1)
λn tn P
Pn
n=1 n
n=1
= O(1)
= O(1)
m−1
n
m
pv pn tv k + O(1)λm tn k
∆λn P
P
n=1
v=1 v
n=1 n
m−1
(3.6)
∆λn Xn + O(1)λm Xm
n=1
= O(1) as m → ∞,
by virtue of (1.6), (1.10), and (2.3). 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 that
m+1
n=2
Pn
pn
k−1
m+1
Tn,2 k = O(1)
n=2
m
= O(1)
pn
Pn Pn−1
n−1
k k
pv λ v t v v=1
1
n−1
Pn−1
v=1
k−1 k m+1
pv λv λv tv pn
P
Pn−1
n
n=v+1
v=1
= O(1)
k−1
pv
(3.7)
m
pv k
λ v tv = O(1) as m → ∞.
Pv
v=1
In view of (2.3), it is clear that
∞
∆λn < ∞,
n=1
(3.8)
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QUASI β-POWER INCREASING SEQUENCES
hence
m+1
n=2
Pn
pn
k−1
m+1
Tn,3 k = O(1)
n=2
m
= O(1)
pn
Pn Pn−1
n−1
k−1
n−1
k
1 Pv ∆λv tv
Pv ∆λv
Pn−1 v=1
v=1
k m+1
Pv tv ∆λv m
pn
∆λv tv k
= O(1)
P
P
n
n−1
n=v+1
v=1
v=1
= O(1)
= O(1)
m−1
v
m
1
1
ti k + O(1)m∆λm tv k
∆ v ∆λv i
v
v=1
v=1
i=1
m−1
m−1
∆λv+1 Xv+1
v ∆2 λv Xv + O(1)
v=1
v=1
+ O(1)m∆λm Xm = O(1) as m → ∞,
(3.9)
by virtue of (1.7), (1.8), (2.2), and (2.3). Since (λn ) is bounded, finally we have that
m+1
n=2
Pn
pn
k−1
m+1
Tn,4 k ≤
n=2
n−1
k k 1
pn
Pv λv+1 tv Pn Pn−1 v=1
v
m
k 1
Pv λv+1 tv v
v=1
m
k
λv+1 tv
= O(1)
v
v=1
= O(1)
1
Pn−1
n−1
Pv
v
v=1
k−1
m+1
pn
P
P
n=v+1 n n−1
(3.10)
m−1
v
m
1
1
tr k + O(1)λm+1 tv k
∆λv+1 = O(1)
r
v
v=1
r =1
v=1
= O(1)
m−1
∆λv+1 Xv+1 + O(1)λm+1 Xm+1
v=1
= O(1) as m → ∞,
by virtue of (1.6), (1.8), (1.11), and (2.3). Therefore, we get that
m Pn k−1 Tn,r k = O(1) as m → ∞, for r = 1, 2, 3, 4.
p
n
n=1
(3.11)
This completes the proof of Theorem 2.1.
Finally, if we take pn = 1 for all values of n in Theorem 2.1, then we get a new result
concerning the |C, 1|k summability factors. Furthermore, if we take pn = 1/(n + 1), then
we get another new result for |N̄, 1/(n + 1)|k summability factors.
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H. BOR AND L. DEBNATH
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
S. Aljančić and D. Arand̄elović, 0-regularly varying functions, Publ. Inst. Math. (Beograd)
(N.S.) 22(36) (1977), 5–22.
H. Bor, On two summability methods, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1,
147–149.
, On |N, pn |k summability factors, Proc. Amer. Math. Soc. 94 (1985), no. 3, 419–422.
, A note on two summability methods, Proc. Amer. Math. Soc. 98 (1986), no. 1, 81–84.
, On absolute summability factors, Proc. Amer. Math. Soc. 118 (1993), no. 1, 71–75.
T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and
Paley, Proc. London Math. Soc. (3) 7 (1957), 113–141.
G. H. Hardy, Divergent Series, Clarendon Press, Oxford University Press, London, 1949.
E. Kogbetliantz, Sur les series absolument sommables par la méthode des moyennes arithmétiques, Bull. Sci. Math. 49 (1925), 234–256 (French).
S. M. Mazhar, On |C, 1k | summability factors of infinite series, Indian J. Math. 14 (1972),
45–48.
, Absolute summability factors of infinite series, Kyungpook Math. J. 39 (1999), no. 1,
67–73.
H. Bor: Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey
E-mail address: bor@erciyes.edu.tr
L. Debnath: Department of Mathematics, University of Texas – Pan American, Edinburg, TX
78539, USA
E-mail address: debnathl@panam.edu