Volume 10 (2009), Issue 2, Article 56, 6 pp.
ON QUASI β-POWER INCREASING SEQUENCES
SANTOSH KR. SAXENA
H. N. 419, JAWAHARPURI , BADAUN
D EPARTMENT OF M ATHEMATICS
T EERTHANKER M AHAVEER U NIVERSITY
M ORADABAD , U.P., I NDIA
ssumath@yahoo.co.in
Received 31 January, 2008; accepted 15 May, 2009
Communicated by S.S. Dragomir
A BSTRACT. In this paper we prove a general theorem on N̄ , pα
n ; δ k summability, which generalizes a theorem of Özarslan [6] on N̄ , pn ; δ k summability, under weaker conditions and by
using quasi β -power increasing sequences instead of almost increasing sequences.
Key words and phrases: Absolute Summability, Summability Factors, Infinite Series.
2000 Mathematics Subject Classification. 40D05, 40F05.
1. I NTRODUCTION
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
(1.1)
holds for all n ≥ m ≥ 1. It should be noted that every almost increasing sequence is a quasi
β-power increasing sequence for any non-negative 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 the theorem of Özarslan [6], replacing an almost increasing sequence by a quasi β-power
increasing
P sequence.
Let
an be a given infinite series
P with partial sums (sn ) and let (pn ) be a sequence with
p0 > 0, pn ≥ 0 for n > 0 and Pn = nν=0 pν . We define
(1.2)
pαn
=
n
X
Aα−1
n−ν pν ,
ν=0
Pnα
=
n
X
pαν ,
α
P−i
= pα−i = 0, i ≥ 1 ,
ν=0
where
(1.3)
Aα0 = 1,
Aαn =
(α + 1) (α + 2) · · · (α + n)
,
n!
(α > −1, n = 1, 2, 3, ...)
The author wishes to express his sincerest thanks to Dr. Rajiv Sinha and the referees for their valuable suggestions for the improvement of
this paper.
035-08
2
S ANTOSH K R . S AXENA
The sequence-to-sequence transformation
n
1 X α
p sν
= α
Pn ν=0 ν
Unα
(1.4)
defines the sequence (Unα ) of the N̄ , pαn mean of the sequence (sn ), generated by the sequence
of coefficientsP(pαn ) (see [7]).
The series an is said to be summable N̄ , pαn k , k ≥ 1, if (see [2])
∞ α k−1
X
Pn
k
α
(1.5)
Unα − Un−1
< ∞,
α
p
n
n=1
and it is said to be summable N̄ , pαn ; δ k , k ≥ 1 and δ ≥ 0, if (see [7])
∞ α δk+k−1
X
Pn
k
α
(1.6)
Unα − Un−1
< ∞.
α
pn
n=1
In the special case when δ = 0, α = 0 (respectively, pn = 1 for all values of n) N̄ , pαn ; δ k
summability is the same as N̄ , pn k (respectively |C, 1; δ|k ) summability.
Mishra and Srivastava [4] proved the following theorem for |C, 1|k summability.
Later on Bor [3] generalized the theorem of Mishra and Srivastava [4] for N̄ , pn k summability.
Quite recently Özarslan [6] has generalized the theorem of Bor [3] under weaker conditions.
For this, Özarslan [6] used the concept of almost increasing sequences. 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]). Obviously every increasing
sequence is an almost increasing sequence but the converse needs not be true as can be seen
n
from the example bn = ne(−1) .
Theorem 1.1. Let (Xn ) be an almost increasing sequence and the sequences (ρn ) and (λn )
such that the conditions
(1.7)
|∆λn | ≤ ρn ,
(1.8)
ρn → 0 as n → ∞,
|λn | Xn = O (1) ,
(1.9)
∞
X
(1.10)
as n → ∞,
n |∆ρn | Xn < ∞.
n=1
are satisfied. If (pn ) is a sequence such that the condition
Pn = O (npn ) ,
(1.11)
as n → ∞,
is satisfied and
(1.12)
δk−1
m X
Pn
n=1
pn
|sn |k = O (Xm ),
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 56, 6 pp.
as m → ∞,
http://jipam.vu.edu.au/
O N Q UASI β-P OWER I NCREASING S EQUENCES
)
( δk−1
∞ δk
X
Pn
1
Pν
1
=O
,
p
P
p
P
n
n−1
ν
ν
n=ν+1
(1.13)
then the series
3
P
an λn is summable N̄ , pn ; δ
k
for k ≥ 1 and 0 ≤ δ < k1 .
2. M AIN R ESULT
The aim of this paper is to generalize Theorem 1.1 for N̄ , pαn ; δ k summability under weaker
conditions by using quasi β-power increasing sequences instead of almost increasing sequences.
Now, we will prove the following theorem.
Theorem 2.1. Let (Xn ) be a quasi β-power increasing sequence for some 0 < β < 1 and the
sequences
(ρn ) and (λn ) such that the conditions (1.7) – (1.10) of Theorem 1.1 are satisfied. If
α
pn is a sequence such that
Pnα = O (npαn ) ,
(2.1)
as n → ∞,
is satisfied and
m α δk−1
X
P
(2.2)
n=1
|sn |k = O (Xm ),
as m → ∞,
( )
∞ α δk−1
δk
X
Pn
1
Pνα
1
=O
,
α
pαn
Pn−1
pαν
Pνα
n=ν+1
(2.3)
then the series
n
α
pn
P
an λn is summable N̄ , pαn ; δ
k
for k ≥ 1 and 0 ≤ δ < k1 .
Remark 1. It may be noted that, if we take (Xn ) as an almost increasing sequence and α = 0
in Theorem 2.1, then we get Theorem 1.1. In this case, conditions (2.1) and (2.2) reduce to
conditions (1.11) and (1.12) respectively and condition (2.3) reduces to (1.13). If additionally
δ = 0, relation (2.3) reduces to
∞
X
pn
1
(2.4)
=O
,
P P
Pν
n=ν+1 n n−1
which always holds.
We need the following lemma for the proof of our theorem.
Lemma 2.2 ([5]). Under the conditions on (Xn ), (βn ) and (λn ) as taken in the statement of the
theorem, the following conditions hold
(2.5)
(2.6)
nρn Xn = O (1) ,
∞
X
as n → ∞,
ρn Xn < ∞.
n=1
P
Proof of Theorem 2.1. Let (Tnα ) be the N̄ , pαn mean of the series an λn . Then by definition,
we have
n
ν
n
1 X αX
1 X α
α
α
Tn = α
pν
aw λ w = α
Pn − Pν−1
aν λ ν .
Pn ν=0 w=0
Pn ν=0
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 56, 6 pp.
http://jipam.vu.edu.au/
4
S ANTOSH K R . S AXENA
Then, for n ≥ 1, we get
Tnα
−
α
Tn−1
n
pαn X α
= α α
P aν λ ν .
Pn Pn−1 ν=1 ν−1
Applying Abel’s transformation, we have
α
Tnα − Tn−1
=
n−1
pαn X
pαn
α
∆
P
λ
s
+
s λ
ν
ν−1 ν
α
α n n
Pnα Pn−1
P
n
ν=1
n−1
n−1
pαn X α
pαn X α
pαn
=− α α
p s ν λν + α α
P sν ∆λν + α sn λn
Pn Pn−1 ν=1 ν
Pn Pn−1 ν=1 ν
Pn
α
α
α
= Tn,1
+ Tn,2
+ Tn,3
,
say.
Since
α
α
α
Tn,1
+ Tn,2
+ Tn,3
k
≤ 3k
α
Tn,1
k
α
+ Tn,2
k
α
+ Tn,3
k
,
to complete the proof of Theorem 2.1, it is sufficient to show that
∞ α δk+k−1
X
P
n=1
n
pαn
α
Tn,w
k
< ∞,
for w = 1, 2, 3.
Now, when k >1, applying
Hölder’s inequality with indices k and k 0 , where
using |λn | = O X1n = O (1), by (1.9), we have
m+1
X
n=2
Pnα
pαn
δk+k−1
α k
Tn,1
=
m+1
X
n=2
≤
m+1
X
Pnα
pαn
δk+k−1
Pnα
pαn
δk−1
n−1
pαn X α
p ν s ν λν
α
Pnα Pn−1
ν=1
1
pαν |sν |k |λν |k
+
1
k0
= 1, and
k
1
n−1
X
α
Pn−1
n=2
ν=1
ν=1
m
m+1
δk−1
X
X Pα
1
k
k
n
α
= O (1)
pν |sν | |λν |
α
pαn
Pn−1
ν=1
n=ν+1
α δk
m
X
Pν
1
k
k
α
= O (1)
pν |sν | |λν |
α
pν
Pνα
ν=1
m α δk−1
X
Pν
= O (1)
|sν |k |λν |k
α
p
ν
ν=1
m
X P α δk−1
ν
= O (1)
|sν |k |λν | |λν |k−1
α
p
ν
ν=1
m
X P α δk−1
ν
= O (1)
|sν |k |λν |
α
p
ν
ν=1
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 56, 6 pp.
α
Pn−1
n−1
X
1
k
!k−1
pαν
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O N Q UASI β-P OWER I NCREASING S EQUENCES
m−1
X
= O (1)
∆ |λν |
ν α δk−1
X
P
ν=1
u=1
m−1
X
= O(1)
u
pαu
k
|su | + O (1) |λm |
5
m α δk−1
X
P
ν
pαν
ν=1
|sν |k
|∆λν | Xν + O (1) |λm | Xm
ν=1
= O (1)
m−1
X
ρν Xν + O (1) |λm | Xm
ν=1
as m → ∞,
= O (1) ,
1
Xν
= O (1) , by
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2. Since νρν = O
α
α
(2.5), using the fact that |∆λn | ≤ ρn by (1.7) and Pn = O (npn ) by (2.1) and after applying the
Hölder’s inequality again, we obtain
m+1
X
n=2
≤
Pnα
pαn
m+1
X
n=2
≤
m+1
X
n=2
= O (1)
= O (1)
δk+k−1
δk−1 Pnα
pαn
Pn
pαn
k (X
n−1
1
α
Pn−1
m
X
ν=1
m X
m−1
X
δk−1
∆ (νρν )
)(
pαν
k
m+1
X
(νρν ) |sν |
1
n−1
X
α
Pn−1
ν=1
k
α δk−1
Pn
pαn
)k−1
pαν
1
α
Pn−1
(νρν )k |sν |k
ν α δk−1
X
P
w=1
m−1
X
Pνα |∆λν | |sν |
ν=1
n=ν+1
Pνα
pαν
)k
ν=1
pαν (νρν )k |sν |k
ν=1
= O (1)
k
α
Pn−1
( n−1
X
1
α δk−1
ν=1
= O (1)
α
Tn,2
w
pαw
k
|sw | + O (1) mρm
m α δk−1
X
P
ν
ν=1
pαν
|sν |k
|∆ (νρν )| Xν + O (1) mρm Xm
ν=1
= O (1)
m−1
X
ν |∆ρν | Xν + O (1)
ν=1
= O (1) ,
m−1
X
ρν+1 Xν+1 + O (1) mρm Xm
ν=1
as m → ∞,
by the virtue of the hypotheses of Theorem 2.1 and Lemma 2.2. Finally, using the fact that
α
Pnα = O (npαn ) , by (2.1) as in Tn,1
, we have
m α δk+k−1
X
P
n=1
n
pαn
α k
Tn,3
= O (1)
m α δk−1
X
P
n
n=1
= O (1) ,
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 56, 6 pp.
pαn
|sn |k |λn |
as m → ∞.
http://jipam.vu.edu.au/
6
S ANTOSH K R . S AXENA
Therefore, we get
∞ α δk+k−1
X
P
n=1
n
pαn
α
Tn,w
k
= O (1) ,
as m → ∞,
for w = 1, 2, 3.
This completes the proof of Theorem 2.1.
If we take pn = 1 and α = 0 for all values of n in Theorem 2.1, then we obtain a result
concerning the |C, 1, δ|k summability.
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[1] S. ALJANČIĆ
(1977), 5–22.
AND
D. ARANDELOVIĆ, O-regular varying functions, Publ. Inst. Math., 22(36)
[2] H. BOR, A note on some absolute summability methods, J. Nigerian Math. Soc., 6 (1987), 41–46.
[3] H. BOR, A note on absolute summability factors, Internet J. Math. and Math. Sci., 17(3) (1994),
479–482.
[4] K.N. MISHRA AND R.S.L. SRIVASTAVA, On absolute Cesàro summability factors of infinite series, Portugal. Math., 42(1) (1985), 53–61.
[5] L. LEINDLER, A new application of quasi power increaseing sequences, Publ. Math. (Debrecen),
58 (2001), 791–796.
[6] H.S. ÖZARSLAN , On almost increasing sequences and its applications, Internat. J. Math. and
Math. Sci, 25(5) (2001), 293–298.
[7] S.K. SAXENA AND S.K. SAXENA, A note on N̄ , pαn ; δ
33(4) (2007), 829–834.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 56, 6 pp.
k
summability factors, Soochow J. Math.,
http://jipam.vu.edu.au/