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On a recent result on absolute summability factors
Applied Mathematics Letters 18 (2005) 1273–1280 www.elsevier.com/locate/aml
On a recent result on absolute summability factors✩ Ekrem Sava¸s∗ Department of Mathematics, Yüzüncü Yil University, 65080 Van, Turkey Received 18 October 2004; accepted 4 February 2005
Abstract In this paper, a general theorem of |A|k summability factors is presented. This generalization is accomplished by replacing the weighted mean matrix in Mazhar’s theorem in [S.M. Mazhar, A remark on a recent result on absolute summability factors, Indian J. Math. 40 (2) (1998) 123–131] with a triangular matrix. © 2005 Published by Elsevier Ltd MSC: 40F05; 40D25 Keywords: Absolute summability; Summability factors
Mazhar [1] obtained sufficient conditions for an λn to be summable |N , pn |k , k ≥ 1. We generalize this result by replacing the weighted mean matrix with a triangular matrix, and by using the correct definition of absolute summability [2]. Let A be a lower triangular matrix and {sn } a sequence. Then An := A series
∞
n
anν sν .
ν=0
an is said to be summable |A|k , k ≥ 1 if
n k−1 |An − An−1 |k < ∞.
(1)
n=1
✩ This
research was completed while the author was a Fulbright scholar at Indiana University, Bloomington, IN, U.S.A., during the fall and spring semesters of 2003–2004. ∗ Fax: +90 432 2251415. E-mail address: [email protected]. 0893-9659/$ - see front matter © 2005 Published by Elsevier Ltd doi:10.1016/j.aml.2005.02.034
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E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
We may associate with A two lower triangular matrices A and Aˆ defined as follows: a¯ nν =
n
anr ,
n, ν = 0, 1, 2, . . . ,
r=ν
and aˆ nν = a¯ nν − a¯ n−1,ν ,
n = 1, 2, 3, . . . .
A positive sequence bn is said to be almost increasing if there exists an increasing sequence {cn } and positive constants A and B such that Acn ≤ bn ≤ Bcn . 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 = e(−1) n. Theorem 1. Let A be a lower triangular matrix non-negative entries satisfying (i) a¯ n0 = 1, n = 0, 1, . . ., (ii) an−1,ν ≥ anν for n ≥ ν + 1, and (iii) nann = O(1). If {X n } is an almost increasing sequence such that (iv) λn → 0 as n → ∞, n X n |2 λn | = O(1), (v) m n=1 |λn | (vi) ∞ n < ∞, and n=1 m 1 1 n (vii) n=1 n |tn |k = O(X m ), where tn := n+1 k=1 kak , then the series an λn is summable |A|k , k ≥ 1. The following lemma is pertinent for the proof of Theorem 1. Lemma 1 ([1]). If {X n } is an almost increasing sequence, under the conditions (iv) and (v) of the theorem we have that (1) ∞ n=1 X n |λn | < ∞, (2) n X n |λn | = O(1), and (3) X n |λn | = O(1). Proof. Let (yn ) be the nth term of the A-transform of ni=0 λi ai . Then, yn := =
n i=0 n ν=0
ani si =
n
ani
i=0
λν a ν
n
ani =
i ν=0 n ν=0
i=ν
λν a ν a¯ nν λν aν
and Yn := yn − yn−1 =
n n (a¯ nν − a¯ n−1,ν )λν aν = aˆ nν λν aν . ν=0
We may write (note that (i) implies that aˆ n0 = 0)
ν=0
(2)
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
Yn =
n aˆ nν λν ν
ν=1 n
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νaν
ν ν−1 aˆ nν λν = r ar − r ar ν ν=1 r=1 r=1 ν n−1 n aˆ nν λν aˆ nn λn = ν r ar + νaν ν n ν=1 ν=1 r=1 =
n−1
(ν aˆ nν )λν
ν=1 n−1
+
n−1 ν +1 ν+1 aˆ n,ν+1 (λν ) tν + tν ν ν ν=1
1 (n + 1)ann λn tn aˆ n,ν+1 λν+1 tν + ν n ν=1
= Tn1 + Tn2 + Tn3 + Tn4 ,
say.
To complete the proof it is sufficient, by Minkowski’s inequality, to show that ∞
n k−1 |Tnr |k < ∞,
r = 1, 2, 3, 4.
for
n=1
From the definition of Aˆ and using (i) and (ii), aˆ n,ν+1 = a¯ n,ν+1 − a¯ n−1,ν+1 n n−1 = ani − an−1,i i=ν+1 ν
= 1−
i=ν+1
ani − 1 +
ν
i=0
=
an−1,i
i=0
ν (an−1,i − an,i ) ≥ 0. i=0
Using Hölder’s inequality and (iii), k n−1 ν + 1 tν n k−1 |Tn1 |k = n k−1 ν aˆ nν λν I1 := ν n=1 n=1 ν=1 k m+1 n−1 = O(1) n k−1 |ν aˆ nν ||λν ||tν | n=1 ν=1 m+1 n−1 = O(1) n k−1 |ν aˆ nν ||λν |k |tν |k m
×
m
n=1 n−1 ν=1
ν=1
k−1
|ν aˆ nν |
.
(3)
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E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
ν aˆ nν = aˆ nν − aˆ n,ν+1 = a¯ nν − a¯ n−1,ν − a¯ n,ν+1 + a¯ n−1,ν+1 = anν − an−1,ν ≤ 0. Thus, using (ii), n−1
n−1 |ν aˆ nν | = (an−1,ν − anν ) = 1 − 1 + ann = ann .
ν=0
ν=0
Using the fact that from (iv) {λn } is bounded, and (vii), I1 = O(1) = O(1)
m+1 n=1 m+1
(nann )k−1
= O(1)
ν=1
= O(1) = O(1)
m ν=1 m
(nann )k−1
= O(1) = O(1) = O(1) = O(1) = O(1)
ν=1 m ν=1 m−1 ν=1 m−1 ν=1 m−1 ν=1
n−1
|λν |k−1 |λν ||ν aˆ nν ||tν |k
ν=1 m+1
|λν ||tν |k
(nann )k−1 |ν aˆ nν |
n=ν+1 m+1
|λν ||tν |
k
|ν aˆ nν |
n=ν+1
|λν |aνν |tν |k
ν=1 m
|λν |k |tν |k |ν aˆ nν |
ν=1
n=1 m
n−1
|λν | |λν |
ν
arr |tr |k −
ν−1
r=1 ν
r=1 m−1
r=1
ν=0
arr |tr |k −
(|λν |) (|λν |)
ν r=1 ν r=1
arr |tr |k
|λν+1 |
ν
arr |tr |k
r=1
arr |tr |k + |λm |
m
arr |tr |k
r=1 m
1 k 1 k |tr | + |λm | |tr | r r r=1
|λν |X ν + O(1)|λm |X m
= O(1),
again using conditions of Lemma 1. Using Hölder’s inequality, I2 :=
m+1 n=2
n k−1 |Tn2 |k =
m+1 n=2
k n−1 ν + 1 tν n k−1 aˆ n,ν+1 (λν ) ν ν=1
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
1277
k ν + 1 |tν | ≤ n k−1 aˆ n,ν+1 |λν | ν n=2 ν=1 k m+1 n−1 = O(1) n k−1 aˆ n,ν+1 |λν ||tν |
m+1
= O(1) ×
n=2 m+1
n−1
n
k−1
ν=1 n−1 ν=1
n=2 n−1
|λν ||tν | aˆ n,ν+1 k
k−1
aˆ n,ν+1 |λν |
.
ν=1
ˆ and conditions (i) and (ii), Using the definition of A, J := = =
n−1
(a¯ n,ν+1 − a¯ n−1,ν+1 )|λν |
ν=1 n−1 ν=1
=
aˆ n,ν+1 |λν |
ν=1 n−1
n
n−1
1−
ν=1
= = =
ani −
i=ν+1
n−1 ν
ν
an−1,i |λν |
i=ν+1
ani − 1 +
i=0
ν
an−1,i |λν |
i=0
(an−1,i − an,i ) |λν |
ν=1 i=0 ν n−1
(an−1,i − an,i )|λν |
ν=1 i=0 n−1 ν=1
ν n−1 (an−1,0 − an,0 )|λν | + (an−1,i − an,i )|λν |
= (an−1,0 − an,0 ) ≤
n−1
n−1
|λν | +
ν=1 n−1
ν (an−1,i − an,i ) i=0
ν=1 i=1 n−1
ν=1
(an−1,i − an,i )
i=1
n−1 ν=i
|λν |
|λν |.
Since {X n } is almost increasing, a condition of the lemma implies that ∞ ν=0 |λν | converges. Therefore |λ | ≤ M and we obtain J ≤ Mann . there exists a positive constant M such that ∞ ν ν=0 We have, using (iii),
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E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280 m+1
I2 := O(1)
(nann )k−1
n=2 m
= O(1)
|λν ||tν |k
ν=1
m+1 n=ν+1
ν=1 m+1
(nann )k−1 aˆ n,ν+1
m+1
|λν ||tν |k
ν=1
From (3),
aˆ n,ν+1 |λν ||tν |k
n=ν+1
m
= O(1)
n−1
aˆ n,ν+1 .
n=ν+1
ν ν m+1 (an−1,i − ani ) = (an−1,i − ani ) i=0
i=0 n=ν+1 ν = (aν,i − am+1,i ) i=0
≤
ν
aν,i = 1.
(4)
i=0
Therefore m
I2 := O(1)
|λν ||tν |k .
ν=1
We may write I2 = O(1)
m
ν|λν |
ν=1
|tν |k . ν
Using summation by parts, and (vi), m−1
I2 := O(1)
ν=1 m−1
= O(1)
(ν|λν |)
ν 1 r=1
r
|tr |k + O(1)m|λm |
m−1 r=1
1 k |tr | r
|(νλν )|X ν + O(1)m|λm |X m .
ν=1
But (νλν ) = νλν − (ν + 1)λν+1 = ν2 λν − λν+1 . Using (v) and properties (1) and (2) of Lemma 1, and the fact that {X n } is almost increasing, I2 = O(1)
m−1 ν=1
= O(1).
ν |2 λν |X ν + O(1)
m−1 ν=1
|λν+1 |X ν+1 + O(1)m|λm |X m
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
1279
Using (iii) and (vii), Hölder’s inequality, summation by parts, properties (1) and (3) of Lemma 1, and (vi), m+1
n
k−1
|Tn3 | = k
n=2
≤ = =
k n−1 1 n aˆ n,ν+1 λν+1 tν ν n=2 ν=1 k m+1 n−1 |λν+1 | k−1 aˆ n,ν+1 |tν | n ν n=2 ν=1 k−1 m+1 n−1 n−1 |λν+1 | k |λν+1 | k−1 O(1) n aˆ n,ν+1 |tν | aˆ n,ν+1 × ν ν n=2 ν=1 ν=1 k−1 m+1 n−1 n−1 |λν+1 | k |λν+1 | k−1 |tν | aˆ n,ν+1 × O(1) (nann ) ν ν n=2 ν=1 ν=1 m+1
k−1
= O(1) = O(1)
m+1
n−1 |λν+1 |
(nann )k−1
ν=1 m+1 k
n=2 m
|λν+1 | |tν | ν ν=1
ν
|tν |k aˆ n,ν+1
(nann )k−1 aˆ n,ν+1
n=ν+1
m
|λν+1 | k m+1 |tν | = O(1) aˆ n,ν+1 ν ν=1 n=ν+1 = O(1) = O(1) = O(1)
m |λν+1 |
ν=1 m−1 ν=1 m−1 ν=1
ν
|tν |k
(|λν+1 |)
ν 1 r=1
r
|tr |k + O(1)|λm+1 |
(|λν+1 |)X ν + O(1)|λm+1 |X m
= O(1).
Finally, using (iii) and (iv), we have m n=1
n
k−1
|Tn4 | = k
m
n
k−1 (n
n=1
= O(1)
m
+ 1)ann λn tn k n
n k−1 |ann |k |λn |k |tn |k
n=1 m = O(1) (nann )k−1 ann |λn |k−1 |λn ||tn |k n=1
m 1 r=1
r
|tr |k
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E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
= O(1)
m
ann |λn ||tn |k
n=1
= O(1), as in the proof of I1 .
Corollary 1. Let { pn } be a positive sequence such that Pn :=
n k=0
pk → ∞, and satisfies
(i) npn = O(Pn ). If {X n } is an almost increasing sequence such that (ii) λn → 0 as n → ∞, (iii) m n X n |2 λn | = O(1), n=1 ∞ |λn | (iv) n=1 n < ∞, and 1 1 n k (v) m O(X m ), where tn := n+1 n=1 n |tn | = k=1 kak , then the series an λn is summable |N , pn |k , k ≥ 1. Proof. Conditions (ii), (iii), (iv) and (v) of Corollary 1 are, respectively, conditions (iv), (v), (vi) and (vii) of Theorem 1. Conditions (i) and (ii) of Theorem 1 are automatically satisfied for any weighted mean method. Condition (iii) of Theorem 1 becomes condition (i) of Corollary 1. References [1] S.M. Mazhar, A remark on a recent result on absolute summability factors, Indian J. Math. 40 (2) (1998) 123–131. [2] B.E. Rhoades, Inclusion theorems for absolute matrix summability methods, J. Math. Anal. Appl. 238 (1999) 82–90.
On a recent result on absolute summability factors✩ Ekrem Sava¸s∗ Department of Mathematics, Yüzüncü Yil University, 65080 Van, Turkey Received 18 October 2004; accepted 4 February 2005
Abstract In this paper, a general theorem of |A|k summability factors is presented. This generalization is accomplished by replacing the weighted mean matrix in Mazhar’s theorem in [S.M. Mazhar, A remark on a recent result on absolute summability factors, Indian J. Math. 40 (2) (1998) 123–131] with a triangular matrix. © 2005 Published by Elsevier Ltd MSC: 40F05; 40D25 Keywords: Absolute summability; Summability factors
Mazhar [1] obtained sufficient conditions for an λn to be summable |N , pn |k , k ≥ 1. We generalize this result by replacing the weighted mean matrix with a triangular matrix, and by using the correct definition of absolute summability [2]. Let A be a lower triangular matrix and {sn } a sequence. Then An := A series
∞
n
anν sν .
ν=0
an is said to be summable |A|k , k ≥ 1 if
n k−1 |An − An−1 |k < ∞.
(1)
n=1
✩ This
research was completed while the author was a Fulbright scholar at Indiana University, Bloomington, IN, U.S.A., during the fall and spring semesters of 2003–2004. ∗ Fax: +90 432 2251415. E-mail address: [email protected]. 0893-9659/$ - see front matter © 2005 Published by Elsevier Ltd doi:10.1016/j.aml.2005.02.034
1274
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
We may associate with A two lower triangular matrices A and Aˆ defined as follows: a¯ nν =
n
anr ,
n, ν = 0, 1, 2, . . . ,
r=ν
and aˆ nν = a¯ nν − a¯ n−1,ν ,
n = 1, 2, 3, . . . .
A positive sequence bn is said to be almost increasing if there exists an increasing sequence {cn } and positive constants A and B such that Acn ≤ bn ≤ Bcn . 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 = e(−1) n. Theorem 1. Let A be a lower triangular matrix non-negative entries satisfying (i) a¯ n0 = 1, n = 0, 1, . . ., (ii) an−1,ν ≥ anν for n ≥ ν + 1, and (iii) nann = O(1). If {X n } is an almost increasing sequence such that (iv) λn → 0 as n → ∞, n X n |2 λn | = O(1), (v) m n=1 |λn | (vi) ∞ n < ∞, and n=1 m 1 1 n (vii) n=1 n |tn |k = O(X m ), where tn := n+1 k=1 kak , then the series an λn is summable |A|k , k ≥ 1. The following lemma is pertinent for the proof of Theorem 1. Lemma 1 ([1]). If {X n } is an almost increasing sequence, under the conditions (iv) and (v) of the theorem we have that (1) ∞ n=1 X n |λn | < ∞, (2) n X n |λn | = O(1), and (3) X n |λn | = O(1). Proof. Let (yn ) be the nth term of the A-transform of ni=0 λi ai . Then, yn := =
n i=0 n ν=0
ani si =
n
ani
i=0
λν a ν
n
ani =
i ν=0 n ν=0
i=ν
λν a ν a¯ nν λν aν
and Yn := yn − yn−1 =
n n (a¯ nν − a¯ n−1,ν )λν aν = aˆ nν λν aν . ν=0
We may write (note that (i) implies that aˆ n0 = 0)
ν=0
(2)
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
Yn =
n aˆ nν λν ν
ν=1 n
1275
νaν
ν ν−1 aˆ nν λν = r ar − r ar ν ν=1 r=1 r=1 ν n−1 n aˆ nν λν aˆ nn λn = ν r ar + νaν ν n ν=1 ν=1 r=1 =
n−1
(ν aˆ nν )λν
ν=1 n−1
+
n−1 ν +1 ν+1 aˆ n,ν+1 (λν ) tν + tν ν ν ν=1
1 (n + 1)ann λn tn aˆ n,ν+1 λν+1 tν + ν n ν=1
= Tn1 + Tn2 + Tn3 + Tn4 ,
say.
To complete the proof it is sufficient, by Minkowski’s inequality, to show that ∞
n k−1 |Tnr |k < ∞,
r = 1, 2, 3, 4.
for
n=1
From the definition of Aˆ and using (i) and (ii), aˆ n,ν+1 = a¯ n,ν+1 − a¯ n−1,ν+1 n n−1 = ani − an−1,i i=ν+1 ν
= 1−
i=ν+1
ani − 1 +
ν
i=0
=
an−1,i
i=0
ν (an−1,i − an,i ) ≥ 0. i=0
Using Hölder’s inequality and (iii), k n−1 ν + 1 tν n k−1 |Tn1 |k = n k−1 ν aˆ nν λν I1 := ν n=1 n=1 ν=1 k m+1 n−1 = O(1) n k−1 |ν aˆ nν ||λν ||tν | n=1 ν=1 m+1 n−1 = O(1) n k−1 |ν aˆ nν ||λν |k |tν |k m
×
m
n=1 n−1 ν=1
ν=1
k−1
|ν aˆ nν |
.
(3)
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E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
ν aˆ nν = aˆ nν − aˆ n,ν+1 = a¯ nν − a¯ n−1,ν − a¯ n,ν+1 + a¯ n−1,ν+1 = anν − an−1,ν ≤ 0. Thus, using (ii), n−1
n−1 |ν aˆ nν | = (an−1,ν − anν ) = 1 − 1 + ann = ann .
ν=0
ν=0
Using the fact that from (iv) {λn } is bounded, and (vii), I1 = O(1) = O(1)
m+1 n=1 m+1
(nann )k−1
= O(1)
ν=1
= O(1) = O(1)
m ν=1 m
(nann )k−1
= O(1) = O(1) = O(1) = O(1) = O(1)
ν=1 m ν=1 m−1 ν=1 m−1 ν=1 m−1 ν=1
n−1
|λν |k−1 |λν ||ν aˆ nν ||tν |k
ν=1 m+1
|λν ||tν |k
(nann )k−1 |ν aˆ nν |
n=ν+1 m+1
|λν ||tν |
k
|ν aˆ nν |
n=ν+1
|λν |aνν |tν |k
ν=1 m
|λν |k |tν |k |ν aˆ nν |
ν=1
n=1 m
n−1
|λν | |λν |
ν
arr |tr |k −
ν−1
r=1 ν
r=1 m−1
r=1
ν=0
arr |tr |k −
(|λν |) (|λν |)
ν r=1 ν r=1
arr |tr |k
|λν+1 |
ν
arr |tr |k
r=1
arr |tr |k + |λm |
m
arr |tr |k
r=1 m
1 k 1 k |tr | + |λm | |tr | r r r=1
|λν |X ν + O(1)|λm |X m
= O(1),
again using conditions of Lemma 1. Using Hölder’s inequality, I2 :=
m+1 n=2
n k−1 |Tn2 |k =
m+1 n=2
k n−1 ν + 1 tν n k−1 aˆ n,ν+1 (λν ) ν ν=1
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
1277
k ν + 1 |tν | ≤ n k−1 aˆ n,ν+1 |λν | ν n=2 ν=1 k m+1 n−1 = O(1) n k−1 aˆ n,ν+1 |λν ||tν |
m+1
= O(1) ×
n=2 m+1
n−1
n
k−1
ν=1 n−1 ν=1
n=2 n−1
|λν ||tν | aˆ n,ν+1 k
k−1
aˆ n,ν+1 |λν |
.
ν=1
ˆ and conditions (i) and (ii), Using the definition of A, J := = =
n−1
(a¯ n,ν+1 − a¯ n−1,ν+1 )|λν |
ν=1 n−1 ν=1
=
aˆ n,ν+1 |λν |
ν=1 n−1
n
n−1
1−
ν=1
= = =
ani −
i=ν+1
n−1 ν
ν
an−1,i |λν |
i=ν+1
ani − 1 +
i=0
ν
an−1,i |λν |
i=0
(an−1,i − an,i ) |λν |
ν=1 i=0 ν n−1
(an−1,i − an,i )|λν |
ν=1 i=0 n−1 ν=1
ν n−1 (an−1,0 − an,0 )|λν | + (an−1,i − an,i )|λν |
= (an−1,0 − an,0 ) ≤
n−1
n−1
|λν | +
ν=1 n−1
ν (an−1,i − an,i ) i=0
ν=1 i=1 n−1
ν=1
(an−1,i − an,i )
i=1
n−1 ν=i
|λν |
|λν |.
Since {X n } is almost increasing, a condition of the lemma implies that ∞ ν=0 |λν | converges. Therefore |λ | ≤ M and we obtain J ≤ Mann . there exists a positive constant M such that ∞ ν ν=0 We have, using (iii),
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E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280 m+1
I2 := O(1)
(nann )k−1
n=2 m
= O(1)
|λν ||tν |k
ν=1
m+1 n=ν+1
ν=1 m+1
(nann )k−1 aˆ n,ν+1
m+1
|λν ||tν |k
ν=1
From (3),
aˆ n,ν+1 |λν ||tν |k
n=ν+1
m
= O(1)
n−1
aˆ n,ν+1 .
n=ν+1
ν ν m+1 (an−1,i − ani ) = (an−1,i − ani ) i=0
i=0 n=ν+1 ν = (aν,i − am+1,i ) i=0
≤
ν
aν,i = 1.
(4)
i=0
Therefore m
I2 := O(1)
|λν ||tν |k .
ν=1
We may write I2 = O(1)
m
ν|λν |
ν=1
|tν |k . ν
Using summation by parts, and (vi), m−1
I2 := O(1)
ν=1 m−1
= O(1)
(ν|λν |)
ν 1 r=1
r
|tr |k + O(1)m|λm |
m−1 r=1
1 k |tr | r
|(νλν )|X ν + O(1)m|λm |X m .
ν=1
But (νλν ) = νλν − (ν + 1)λν+1 = ν2 λν − λν+1 . Using (v) and properties (1) and (2) of Lemma 1, and the fact that {X n } is almost increasing, I2 = O(1)
m−1 ν=1
= O(1).
ν |2 λν |X ν + O(1)
m−1 ν=1
|λν+1 |X ν+1 + O(1)m|λm |X m
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
1279
Using (iii) and (vii), Hölder’s inequality, summation by parts, properties (1) and (3) of Lemma 1, and (vi), m+1
n
k−1
|Tn3 | = k
n=2
≤ = =
k n−1 1 n aˆ n,ν+1 λν+1 tν ν n=2 ν=1 k m+1 n−1 |λν+1 | k−1 aˆ n,ν+1 |tν | n ν n=2 ν=1 k−1 m+1 n−1 n−1 |λν+1 | k |λν+1 | k−1 O(1) n aˆ n,ν+1 |tν | aˆ n,ν+1 × ν ν n=2 ν=1 ν=1 k−1 m+1 n−1 n−1 |λν+1 | k |λν+1 | k−1 |tν | aˆ n,ν+1 × O(1) (nann ) ν ν n=2 ν=1 ν=1 m+1
k−1
= O(1) = O(1)
m+1
n−1 |λν+1 |
(nann )k−1
ν=1 m+1 k
n=2 m
|λν+1 | |tν | ν ν=1
ν
|tν |k aˆ n,ν+1
(nann )k−1 aˆ n,ν+1
n=ν+1
m
|λν+1 | k m+1 |tν | = O(1) aˆ n,ν+1 ν ν=1 n=ν+1 = O(1) = O(1) = O(1)
m |λν+1 |
ν=1 m−1 ν=1 m−1 ν=1
ν
|tν |k
(|λν+1 |)
ν 1 r=1
r
|tr |k + O(1)|λm+1 |
(|λν+1 |)X ν + O(1)|λm+1 |X m
= O(1).
Finally, using (iii) and (iv), we have m n=1
n
k−1
|Tn4 | = k
m
n
k−1 (n
n=1
= O(1)
m
+ 1)ann λn tn k n
n k−1 |ann |k |λn |k |tn |k
n=1 m = O(1) (nann )k−1 ann |λn |k−1 |λn ||tn |k n=1
m 1 r=1
r
|tr |k
1280
E. Sava¸s / Applied Mathematics Letters 18 (2005) 1273–1280
= O(1)
m
ann |λn ||tn |k
n=1
= O(1), as in the proof of I1 .
Corollary 1. Let { pn } be a positive sequence such that Pn :=
n k=0
pk → ∞, and satisfies
(i) npn = O(Pn ). If {X n } is an almost increasing sequence such that (ii) λn → 0 as n → ∞, (iii) m n X n |2 λn | = O(1), n=1 ∞ |λn | (iv) n=1 n < ∞, and 1 1 n k (v) m O(X m ), where tn := n+1 n=1 n |tn | = k=1 kak , then the series an λn is summable |N , pn |k , k ≥ 1. Proof. Conditions (ii), (iii), (iv) and (v) of Corollary 1 are, respectively, conditions (iv), (v), (vi) and (vii) of Theorem 1. Conditions (i) and (ii) of Theorem 1 are automatically satisfied for any weighted mean method. Condition (iii) of Theorem 1 becomes condition (i) of Corollary 1. References [1] S.M. Mazhar, A remark on a recent result on absolute summability factors, Indian J. Math. 40 (2) (1998) 123–131. [2] B.E. Rhoades, Inclusion theorems for absolute matrix summability methods, J. Math. Anal. Appl. 238 (1999) 82–90.