ON QUASI β-POWER INCREASING SEQUENCES
SANTOSH Kr. SAXENA
H. N. 419, Jawaharpuri, Badaun
Department of Mathematics
Teerthanker Mahaveer University
Moradabad, U.P., India
EMail: ssumath@yahoo.co.in
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
vol. 10, iss. 2, art. 56, 2009
Title Page
Contents
Received:
31 January, 2008
Accepted:
15 May, 2009
Communicated by:
S.S. Dragomir
2000 AMS Sub. Class.:
40D05, 40F05.
Key words:
Absolute Summability, Summability Factors, Infinite Series.
Abstract:
In this paper we prove a general theorem on N̄ , pα
; δ summability, which
n k
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.
Acknowledgements:
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.
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Contents
1
Introduction
3
2
Main Result
6
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
vol. 10, iss. 2, art. 56, 2009
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1.
Introduction
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
P sequence by a quasi β-power increasing sequence.
Let
an be a given infinite series with partial
Pnsums (sn ) and let (pn ) be a sequence with p0 > 0, pn ≥ 0 for n > 0 and Pn = ν=0 pν . We define
(1.2)
pαn =
n
X
Aα−1
n−ν pν ,
ν=0
Pnα =
n
X
pαν ,
α
= pα−i = 0, i ≥ 1 ,
P−i
ν=0
where
(1.3)
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
vol. 10, iss. 2, art. 56, 2009
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Aα0 = 1,
Aαn =
(α + 1) (α + 2) · · · (α + n)
, (α > −1, n = 1, 2, 3, ...)
n!
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The sequence-to-sequence transformation
(1.4)
n
1 X α
α
p sν
Un = α
Pn ν=0 ν
defines the sequence (Unα ) of the N̄ , pαn mean of the sequence (sn ), generated by
the sequence of coefficients (pαn ) (see [7]).
Close
The series
P
an is said to be summable N̄ , pαn k , k ≥ 1, if (see [2])
∞ α k−1
X
α
Pn
α k
Un − Un−1
(1.5)
< ∞,
pαn
n=1
and it is said to be summable N̄ , pαn ; δ k , k ≥ 1 and δ ≥ 0, if (see [7])
(1.6)
∞ X
n=1
Pnα
pαn
δk+k−1
α
α k
Un − Un−1
< ∞.
case when δ = 0, α = 0 (respectively,
pn = 1 for all values of n)
In theα special
N̄ , pn ; δ summability is the same as N̄ , pn (respectively |C, 1; δ| ) summability.
k
k
k
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
n
but the converse needs not be true as can be seen 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)
(1.8)
(1.9)
|∆λn | ≤ ρn ,
ρn → 0 as n → ∞,
|λn | Xn = O (1) , as n → ∞,
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
vol. 10, iss. 2, art. 56, 2009
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∞
X
(1.10)
n |∆ρn | Xn < ∞.
n=1
are satisfied. If (pn ) is a sequence such that the condition
Pn = O (npn ) ,
(1.11)
as n → ∞,
Quasi β-Power Increasing
Sequences
is satisfied and
δk−1
m X
Pn
(1.12)
n=1
pn
Santosh Kr. Saxena
|sn |k = O (Xm ),
as m → ∞,
vol. 10, iss. 2, art. 56, 2009
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∞
X
(1.13)
n=ν+1
then the series
P
Pn
pn
δk−1
1
Pn−1
(
=O
Pν
pν
δk
)
1
,
Pν
an λn is summable N̄ , pn ; δ k for k ≥ 1 and 0 ≤ δ < k1 .
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2.
Main Result
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 pαn is a sequence such that
Pnα
(2.1)
=
O (npαn ) ,
as n → ∞,
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
vol. 10, iss. 2, art. 56, 2009
is satisfied and
(2.2)
m α δk−1
X
P
n=1
n
α
pn
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k
|sn | = O (Xm ),
as m → ∞,
( )
∞ α δk−1
δk
X
Pn
1
Pνα
1
(2.3)
=O
,
α
α
α
α
p
P
p
P
n−1
n
ν
ν
n=ν+1
P
then the series 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
P
n
n−1
ν
n=ν+1
which always holds.
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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
nρn Xn = O (1) ,
(2.5)
∞
X
(2.6)
as n → ∞,
Quasi β-Power Increasing
Sequences
ρn Xn < ∞.
Santosh Kr. Saxena
n=1
Proof of Theorem 2.1. Let
definition, we have
(Tnα )
be the
vol. 10, iss. 2, art. 56, 2009
N̄ , pαn
mean of the series
P
an λn . Then by
Title Page
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
Then, for n ≥ 1, we get
α
Tnα − 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
pα
α
= α α
∆ Pν−1
λν sν + nα sn λn
Pn Pn−1 ν=1
Pn
n−1
n−1
pαn X α
pαn X α
pα
p ν sν λν + α α
Pν sν ∆λν + nα sn λn
=− α α
Pn Pn−1 ν=1
Pn Pn−1 ν=1
Pn
α
α
α
= Tn,1
+ Tn,2
+ Tn,3
,
say.
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Since
α
α k α k α
α k
α k
Tn,1 + Tn,2
+ Tn,3 ,
+ Tn,3
≤ 3k Tn,1
+ Tn,2
to complete the proof of Theorem 2.1, it is sufficient to show that
∞ α δk+k−1
X
α k
Pn
Tn,w < ∞, for w = 1, 2, 3.
α
p
n
n=1
Quasi β-Power Increasing
Sequences
1
1
0
Now, when k > 1, applying
Hölder’s inequality with indices k and k , where k + k0 =
1, and using |λn | = O
m+1
X
n=2
Pnα
pαn
δk+k−1
1
Xn
= O (1), by (1.9), we have
X
α k m+1
Tn,1 =
n=2
≤
m+1
X
n=2
= O (1)
Pnα
pαn
Pnα
pαn
m
X
k
δk+k−1 n−1
α
X
p
n
α
p
s
λ
α α
ν ν
Pn Pn−1 ν=1 ν
δk−1
pαν
ν=1
= O (1)
= O (1)
m
X
ν=1
m X
ν=1
= O (1)
pαν
1
pαν
k
Pνα
pαν
δk−1
ν
pαν
|sν |k |λν |k
|sν |k |λν | |λν |k−1
Contents
n−1
X
α
Pn−1
ν=1
vol. 10, iss. 2, art. 56, 2009
Title Page
1
k
|sν | |λν |
α
Pn−1
ν=1
m+1
X P α δk−1
k
k
n
|sν | |λν |
α
p
n
n=ν+1
α δk
Pν
1
|sν |k |λν |k
α
pν
Pνα
m α δk−1
X
P
ν=1
n−1
X
Santosh Kr. Saxena
1
α
Pn−1
!k−1
pαν
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= O (1)
= O (1)
= O(1)
= O (1)
m α δk−1
X
P
ν=1
m−1
X
ν=1
m−1
X
ν
pαν
∆ |λν |
|sν |k |λν |
ν α δk−1
X
P
u=1
u
pαu
k
|su | + O (1) |λm |
m α δk−1
X
P
ν
ν=1
pαν
|sν |k
|∆λν | Xν + O (1) |λm | Xm
ν=1
m−1
X
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
vol. 10, iss. 2, art. 56, 2009
ρν Xν + O (1) |λm | Xm
ν=1
= O (1) ,
as m → ∞,
by virtue of the hypotheses of Theorem 2.1 and Lemma 2.2. Since νρν = O X1ν =
O (1) , by (2.5), using the fact that |∆λn | ≤ ρn by (1.7) and Pnα = O (npαn ) by (2.1)
and after applying the Hölder’s inequality again, we obtain
m+1
X P α δk+k−1 n
α k
Tn,2
pαn
n=2
( n−1
)k
m+1
X P α δk−1 1 k X
n
≤
Pνα |∆λν | |sν |
α
α
p
P
n−1
n
n=2
ν=1
( n−1
)(
)k−1
n−1
m+1
X P α δk−1 1
X
X
1
n
≤
pαν (νρν )k |sν |k
pαν
α
α
α
p
P
P
n−1
n−1 ν=1
n
n=2
ν=1
m
m+1
δk−1
X
X Pα
1
n
= O (1)
pαν (νρν )k |sν |k
α
α
pn
Pn−1
ν=1
n=ν+1
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= O (1)
= O (1)
= O (1)
= O (1)
m α δk−1
X
P
ν
ν=1
m−1
X
ν=1
m−1
X
ν=1
m−1
X
pαν
∆ (νρν )
ν α δk−1
X
P
w=1
w
α
pw
k
|sw | + O (1) mρm
m α δk−1
X
P
ν=1
ν
α
pν
|sν |k
|∆ (νρν )| Xν + O (1) mρm Xm
ν |∆ρν | Xν + O (1)
ν=1
= O (1) ,
(νρν )k |sν |k
m−1
X
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
ρν+1 Xν+1 + O (1) mρm Xm
vol. 10, iss. 2, art. 56, 2009
ν=1
as m → ∞,
Title Page
by the virtue of the hypotheses of Theorem 2.1 and Lemma 2.2. Finally, using the
α
fact that Pnα = O (npαn ) , by (2.1) as in Tn,1
, we have
m α δk−1
m α δk+k−1
X
X
α k
Pn
Pn
Tn,3 = O (1)
|sn |k |λn |
α
α
pn
pn
n=1
n=1
= O (1) ,
as m → ∞.
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Therefore, we get
∞ α δk+k−1
X
α k
Pn
Tn,w = O (1) ,
α
p
n
n=1
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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.
Close
References
[1] S. ALJANČIĆ AND D. ARANDELOVIĆ, O-regular varying functions, Publ.
Inst. Math., 22(36) (1977), 5–22.
[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.
Quasi β-Power Increasing
Sequences
Santosh Kr. Saxena
vol. 10, iss. 2, art. 56, 2009
[4] K.N. MISHRA AND R.S.L. SRIVASTAVA, On absolute Cesàro summability
factors of infinite series, Portugal. Math., 42(1) (1985), 53–61.
Title Page
[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 k summability factors,
Soochow J. Math., 33(4) (2007), 829–834.
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