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Factors for generalized absolute Cesàro summability
Mathematical and Computer Modelling 53 (2011) 1150–1153
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Factors for generalized absolute Cesàro summability Hüseyin Bor P.O.Box 121, 06502 Bahçelievler, Ankara, Turkey
article
abstract
info
In this paper, a known theorem dealing with |C , α|k summability factors has been generalized for |C , α, β|k summability factors. © 2010 Elsevier Ltd. All rights reserved.
Article history: Received 17 August 2010 Accepted 24 November 2010 Keywords: Cesàro summability Almost increasing sequences Summability factors
1. Introduction 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]). Let an be a given infinite series with partial sums (sn ). We denote α,β
α,β
the nth Cesàro means of order (α, β), with α + β > −1, of the sequence (sn ) and (nan ), respectively, i.e.,
by un and tn (see [2]) uα,β = n tnα,β =
1 α+β
An
1 α+β
An
n −
1 β Aα− n−v Av sv
(1)
1 β Aα− n−v Av v av ,
(2)
v=0 n −
v=1
where Anα+β = O(nα+β ), The series ∞ −
∑
α + β > −1 ,
α+β
A0
α+β
= 1 and A−n = 0 for n > 0.
(3)
an is said to be summable |C , α, β|k , k ≥ 1 and α + β > −1, if (see [3]) α,β
nk−1 |uα,β − un−1 |k < ∞. n
(4)
n =1
α,β
Since tn
α,β
= n(un
∞ − 1 n =1
n
α,β
− un−1 ) (see [3]), condition (4) can also be written as
|tnα,β |k < ∞.
(5)
If we take β = 0, then |C , α, β|k summability reduces to |C , α|k (see [4]) summability. It should be also noted that obviously the (C , α, 0) mean is the same as the (C , α) mean.
E-mail addresses: [email protected], [email protected]. 0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.11.081
H. Bor / Mathematical and Computer Modelling 53 (2011) 1150–1153
1151
Bor and Srivastava [5] have proved the following theorem dealing with |C , α|k summability factors of infinite series. Theorem A. Let (Xn ) be an almost increasing sequence and let there be sequences (βn ) and (λn ) such that
|1λn | ≤ βn
(6)
βn → 0 as n → ∞
(7)
∞ −
n|1βn |Xn < ∞
(8)
n =1
|λn |Xn = O(1) as n → ∞.
(9)
If the sequence (θnα ) defined by (see [6])
θnα = |tnα |,
α=1
α
θn = max
1≤v≤n
|tvα |,
(10)
0<α<1
(11)
satisfies the condition m − 1
n n =1
(θnα )k = O(Xm ) as m → ∞,
then the series
∑
(12)
an λn is summable |C , α|k , k ≥ 1 and 0 < α ≤ 1.
2. The main result The aim of this paper is to generalize Theorem A for |C , α, β|k summability. We shall prove the following theorem. Theorem. Let (Xn ) be an almost increasing sequence and let there be sequences (βn ) and (λn ) such that conditions (6)–(9) α,β of Theorem A are satisfied. If the sequence (θn ) defined by
θnα,β = |tnα,β |,
α = 1, β > −1
θnα,β = max |tvα,β |, 1≤v≤n
0 < α < 1, β > −1
(13) (14)
satisfies the condition m − 1
n n =1
(θnα,β )k = O(Xm ) as m → ∞,
(15)
then the series an λn is summable |C , α, β|k for 0 < α ≤ 1, β > −1, α + β > 0 and k ≥ 1. Since |C , α, β|k and |C , α|k summabilities are different from each other, they have different summability fields. Therefore, it is clear that the |C , α, β|k summability is more general than the |C , α|k summability. Thus, for β = 0 the |C , α|k summability is a special case of the |C , α, β|k summability. For this reason, if we take β = 0, then we get Theorem A. In this case, the conditions (13)–(15) are reduced to conditions (10)–(12), respectively.
∑
We need the following lemmas for the proof of our theorem. Lemma 1 ([7]). If (Xn ) is an almost increasing sequence, then under the conditions (6)–(9) of Theorem A, we have that nXn βn = O(1), ∞ −
βn Xn < ∞.
(16) (17)
n =1
Lemma 2 ([8]). If 0 < α ≤ 1, β > −1 and 1 ≤ v ≤ n, then
v m − − α−1 β α−1 β An−p Ap ap ≤ max Am−p Ap ap . p=0 1≤m≤v p=0
(18)
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H. Bor / Mathematical and Computer Modelling 53 (2011) 1150–1153
3. Proof of the theorem α,β
Let (Tn
) be the nth (C , α, β) mean of the sequence (nan λn ). Then, by (2), we have
Tnα,β =
n −
1 α+β
An
v=1
1 β Aα− n−v Av v av λv .
First applying Abel’s transformation and then using Lemma 2, we have that Tnα,β =
n−1 −
1 α+β
An
|Tn | ≤ ≤
v=1
1
α,β
1λv
α+β
An
1
v −
1 β Aα− n−p Ap pap +
p=1
α+β
An
v=1
1 β Anα− −v Av v av ,
v − |λn | α−1 β |1λv | An−p Ap pap + α+β A n v=1 p=1
n−1 −
n−1 −
Aαv Aβv θvα,β |1λv | v=1 α,β α,β Tn,1 + Tn,2 , say. α+β
An
=
n λn −
n − α−1 β An−v Av v av v=1
+ |λn |θnα,β
Since α,β
α,β
α,β
α,β
|Tn,1 + Tn,2 |k ≤ 2k (|Tn,1 |k + |Tn,2 |k ), in order to complete the proof of the theorem, by (5), it is sufficient to show that ∞ − 1
n n =1
k |Tnα,β ,r | < ∞,
for r = 1, 2.
Whenever k > 1, we can apply Hölder’s inequality with indices k and k′ ; where m+1
−1 n=2
n
α,β |Tn,1 |k
k n −1 − 1 1 − α β α,β Av Av θv 1λv ≤ α+β n An v=1 n =2 k−1 m +1 n −1 n −1 − − − 1 = O(1) v αk v β k βv (θvα,β )k × βv 1+(α+β)k m+1
n =2
= O(1)
m −
v=1
= O(1)
m −
v=1
= O(1)
m −
v=1
n
v=1
v (α+β)k βv (θvα,β )k v (α+β)k βv (θvα,β )k
= O(1)
−
v=1
m+1
∆(vβv )
v=1
v − 1 p=1
p
n1+(α+β)k n=v+1
−
∞
∫ v
dx x1+(α+β)k
m −
1
vβv (θvα,β )k v v=1
(θpα,β )k + O(1)mβm
m−1
= O(1)
1
−
βv (θvα,β )k = O(1)
m−1
m − 1 α,β k (θv ) v v=1
m−1
v|1βv |Xv + O(1)
v=1
−
βv Xv + O(1)mβm Xm
v=1
= O(1) as m → ∞, in view of the hypotheses of the theorem and Lemma 1. Similarly, we have that m − 1
n n =1
1 k
α,β
|Tn,2 |k = O(1)
m − |λn | n =1
n
(θnα,β )k
m−1
= O(1)
− n=1
∆|λn |
n m − − 1 α,β k 1 α,β k (θv ) + O(1)|λm | (θv ) v v v=1 v=1
+
1 k′
= 1, we get that
H. Bor / Mathematical and Computer Modelling 53 (2011) 1150–1153 m−1
= O(1)
−
|1λn |Xn + O(1)|λm |Xm
n =1 m−1
= O(1)
−
βn Xn + O(1)|λm |Xm
n =1
= O(1) as m → ∞. Therefore, by using (5) we get that ∞ − 1
n n =1
k |Tnα,β ,r | < ∞,
for r = 1, 2.
This completes the proof of the theorem. References [1] [2] [3] [4] [5] [6] [7] [8]
S. Aljancic, D. Arandelovic, O-regularly varying functions, Publ. Inst. Math. 22 (1977) 5–22. D. Borwein, Theorems on some methods of summability, Quart. J. Math. Oxford Ser. 9 (1958) 310–316. G. Das, A Tauberian theorem for absolute summability, Proc. Cambridge Philos. Soc. 67 (1970) 321–326. T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. Lond. Math. Soc. 7 (1957) 113–141. H. Bor, H.M. Srivastava, Almost increasing sequences and their applications, Int. J. Pure Appl. Math. 3 (2002) 29–35. T. Pati, The summability factors of infinite series, Duke Math. J. 21 (1954) 271–284. S.M. Mazhar, A note on absolute summability factors, Bull. Inst. Math. Acad. Sin. 25 (1997) 233–242. H. Bor, On a new application of quasi power increasing sequences, Proc. Estonian Acad. Sci. Phys. Math. 57 (2008) 205–209.
1153
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Factors for generalized absolute Cesàro summability Hüseyin Bor P.O.Box 121, 06502 Bahçelievler, Ankara, Turkey
article
abstract
info
In this paper, a known theorem dealing with |C , α|k summability factors has been generalized for |C , α, β|k summability factors. © 2010 Elsevier Ltd. All rights reserved.
Article history: Received 17 August 2010 Accepted 24 November 2010 Keywords: Cesàro summability Almost increasing sequences Summability factors
1. Introduction 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]). Let an be a given infinite series with partial sums (sn ). We denote α,β
α,β
the nth Cesàro means of order (α, β), with α + β > −1, of the sequence (sn ) and (nan ), respectively, i.e.,
by un and tn (see [2]) uα,β = n tnα,β =
1 α+β
An
1 α+β
An
n −
1 β Aα− n−v Av sv
(1)
1 β Aα− n−v Av v av ,
(2)
v=0 n −
v=1
where Anα+β = O(nα+β ), The series ∞ −
∑
α + β > −1 ,
α+β
A0
α+β
= 1 and A−n = 0 for n > 0.
(3)
an is said to be summable |C , α, β|k , k ≥ 1 and α + β > −1, if (see [3]) α,β
nk−1 |uα,β − un−1 |k < ∞. n
(4)
n =1
α,β
Since tn
α,β
= n(un
∞ − 1 n =1
n
α,β
− un−1 ) (see [3]), condition (4) can also be written as
|tnα,β |k < ∞.
(5)
If we take β = 0, then |C , α, β|k summability reduces to |C , α|k (see [4]) summability. It should be also noted that obviously the (C , α, 0) mean is the same as the (C , α) mean.
E-mail addresses: [email protected], [email protected]. 0895-7177/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2010.11.081
H. Bor / Mathematical and Computer Modelling 53 (2011) 1150–1153
1151
Bor and Srivastava [5] have proved the following theorem dealing with |C , α|k summability factors of infinite series. Theorem A. Let (Xn ) be an almost increasing sequence and let there be sequences (βn ) and (λn ) such that
|1λn | ≤ βn
(6)
βn → 0 as n → ∞
(7)
∞ −
n|1βn |Xn < ∞
(8)
n =1
|λn |Xn = O(1) as n → ∞.
(9)
If the sequence (θnα ) defined by (see [6])
θnα = |tnα |,
α=1
α
θn = max
1≤v≤n
|tvα |,
(10)
0<α<1
(11)
satisfies the condition m − 1
n n =1
(θnα )k = O(Xm ) as m → ∞,
then the series
∑
(12)
an λn is summable |C , α|k , k ≥ 1 and 0 < α ≤ 1.
2. The main result The aim of this paper is to generalize Theorem A for |C , α, β|k summability. We shall prove the following theorem. Theorem. Let (Xn ) be an almost increasing sequence and let there be sequences (βn ) and (λn ) such that conditions (6)–(9) α,β of Theorem A are satisfied. If the sequence (θn ) defined by
θnα,β = |tnα,β |,
α = 1, β > −1
θnα,β = max |tvα,β |, 1≤v≤n
0 < α < 1, β > −1
(13) (14)
satisfies the condition m − 1
n n =1
(θnα,β )k = O(Xm ) as m → ∞,
(15)
then the series an λn is summable |C , α, β|k for 0 < α ≤ 1, β > −1, α + β > 0 and k ≥ 1. Since |C , α, β|k and |C , α|k summabilities are different from each other, they have different summability fields. Therefore, it is clear that the |C , α, β|k summability is more general than the |C , α|k summability. Thus, for β = 0 the |C , α|k summability is a special case of the |C , α, β|k summability. For this reason, if we take β = 0, then we get Theorem A. In this case, the conditions (13)–(15) are reduced to conditions (10)–(12), respectively.
∑
We need the following lemmas for the proof of our theorem. Lemma 1 ([7]). If (Xn ) is an almost increasing sequence, then under the conditions (6)–(9) of Theorem A, we have that nXn βn = O(1), ∞ −
βn Xn < ∞.
(16) (17)
n =1
Lemma 2 ([8]). If 0 < α ≤ 1, β > −1 and 1 ≤ v ≤ n, then
v m − − α−1 β α−1 β An−p Ap ap ≤ max Am−p Ap ap . p=0 1≤m≤v p=0
(18)
1152
H. Bor / Mathematical and Computer Modelling 53 (2011) 1150–1153
3. Proof of the theorem α,β
Let (Tn
) be the nth (C , α, β) mean of the sequence (nan λn ). Then, by (2), we have
Tnα,β =
n −
1 α+β
An
v=1
1 β Aα− n−v Av v av λv .
First applying Abel’s transformation and then using Lemma 2, we have that Tnα,β =
n−1 −
1 α+β
An
|Tn | ≤ ≤
v=1
1
α,β
1λv
α+β
An
1
v −
1 β Aα− n−p Ap pap +
p=1
α+β
An
v=1
1 β Anα− −v Av v av ,
v − |λn | α−1 β |1λv | An−p Ap pap + α+β A n v=1 p=1
n−1 −
n−1 −
Aαv Aβv θvα,β |1λv | v=1 α,β α,β Tn,1 + Tn,2 , say. α+β
An
=
n λn −
n − α−1 β An−v Av v av v=1
+ |λn |θnα,β
Since α,β
α,β
α,β
α,β
|Tn,1 + Tn,2 |k ≤ 2k (|Tn,1 |k + |Tn,2 |k ), in order to complete the proof of the theorem, by (5), it is sufficient to show that ∞ − 1
n n =1
k |Tnα,β ,r | < ∞,
for r = 1, 2.
Whenever k > 1, we can apply Hölder’s inequality with indices k and k′ ; where m+1
−1 n=2
n
α,β |Tn,1 |k
k n −1 − 1 1 − α β α,β Av Av θv 1λv ≤ α+β n An v=1 n =2 k−1 m +1 n −1 n −1 − − − 1 = O(1) v αk v β k βv (θvα,β )k × βv 1+(α+β)k m+1
n =2
= O(1)
m −
v=1
= O(1)
m −
v=1
= O(1)
m −
v=1
n
v=1
v (α+β)k βv (θvα,β )k v (α+β)k βv (θvα,β )k
= O(1)
−
v=1
m+1
∆(vβv )
v=1
v − 1 p=1
p
n1+(α+β)k n=v+1
−
∞
∫ v
dx x1+(α+β)k
m −
1
vβv (θvα,β )k v v=1
(θpα,β )k + O(1)mβm
m−1
= O(1)
1
−
βv (θvα,β )k = O(1)
m−1
m − 1 α,β k (θv ) v v=1
m−1
v|1βv |Xv + O(1)
v=1
−
βv Xv + O(1)mβm Xm
v=1
= O(1) as m → ∞, in view of the hypotheses of the theorem and Lemma 1. Similarly, we have that m − 1
n n =1
1 k
α,β
|Tn,2 |k = O(1)
m − |λn | n =1
n
(θnα,β )k
m−1
= O(1)
− n=1
∆|λn |
n m − − 1 α,β k 1 α,β k (θv ) + O(1)|λm | (θv ) v v v=1 v=1
+
1 k′
= 1, we get that
H. Bor / Mathematical and Computer Modelling 53 (2011) 1150–1153 m−1
= O(1)
−
|1λn |Xn + O(1)|λm |Xm
n =1 m−1
= O(1)
−
βn Xn + O(1)|λm |Xm
n =1
= O(1) as m → ∞. Therefore, by using (5) we get that ∞ − 1
n n =1
k |Tnα,β ,r | < ∞,
for r = 1, 2.
This completes the proof of the theorem. References [1] [2] [3] [4] [5] [6] [7] [8]
S. Aljancic, D. Arandelovic, O-regularly varying functions, Publ. Inst. Math. 22 (1977) 5–22. D. Borwein, Theorems on some methods of summability, Quart. J. Math. Oxford Ser. 9 (1958) 310–316. G. Das, A Tauberian theorem for absolute summability, Proc. Cambridge Philos. Soc. 67 (1970) 321–326. T.M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. Lond. Math. Soc. 7 (1957) 113–141. H. Bor, H.M. Srivastava, Almost increasing sequences and their applications, Int. J. Pure Appl. Math. 3 (2002) 29–35. T. Pati, The summability factors of infinite series, Duke Math. J. 21 (1954) 271–284. S.M. Mazhar, A note on absolute summability factors, Bull. Inst. Math. Acad. Sin. 25 (1997) 233–242. H. Bor, On a new application of quasi power increasing sequences, Proc. Estonian Acad. Sci. Phys. Math. 57 (2008) 205–209.
1153