Rhoades 3 1 Department of Mathematics, Istanbul Commerce University, 34672 ¨ Usk ¨udar, Istanbul, Turkey 2 Department of Mathematics, Faculty of Arts & Sciences, Y ¨uz ¨unc ¨u Yil Univer
Trang 1Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 257318, 4 pages
doi:10.1155/2008/257318
Research Article
On the Ces ´aro Summability of Double Series
E Savas¸, 1 H S¸evli, 2 and B E Rhoades 3
1 Department of Mathematics, Istanbul Commerce University, 34672 ¨ Usk ¨udar, Istanbul, Turkey
2 Department of Mathematics, Faculty of Arts & Sciences, Y ¨uz ¨unc ¨u Yil University, 65080 Van, Turkey
3 Department of Mathematics, Indiana University, Bloomington, IN 47405, USA
Correspondence should be addressed to E Savas¸, ekremsavas@yahoo.com
Received 18 July 2007; Accepted 19 August 2007
Recommended by Martin J Bohner
In a recent paper by Savas¸ and S¸evli2007, it was shown that each Ces´aro matrix of order α, for
α > −1, is absolutely kth power conservative for k ≥ 1 In this paper we extend this result to double
Ces´aro matrices.
Copyright q 2008 E Savas¸ et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The concept of absolute summability of order k ≥ 1 was defined by Flett 1 as follows Let
a kbe a series with partial sumss n , A an infinite matrix Thena kis said to be absolutely summableA of order k ≥ 1 if
∞
n1
n k−1 T n−1 − T nk < ∞, 1 where
T n:∞
k0
Denote byAk the sequence space defined by
Ak
s n:
∞
n1
n k−1 a nk < ∞; a n s n − s n−1
3 for k ≥ 1 A matrix T is said to be a bounded linear operator on A k, written T ∈ BA k, if
T : A k → Ak In 1970, Das2 defined such a matrix to be absolutely kth power conservative
Trang 22 Journal of Inequalities and Applications
fork ≥ 1 In that paper, he proved that every conservative Hausdorff matrix H ∈ BA k for
k ≥ 1 In a recent paper 3, the first two authors proved every Ces´aro matrix of order α, for
α > −1, C, α ∈ BA k for k ≥ 1 Since the Ces´aro matrices of order α for −1 < α < 0 are not
conservative, their result shows that being conservative is not a necessary condition for being absolutelykth power conservative.
In this paper, we extend the result of3 to double summability, thereby demonstrating that the property of being conservative is again not necessary for doubly infinite matrices to be absolutelykth power conservative.
Let∞
m0∞
n0 a mnbe an infinite double series with real or complex numbers, with partial sums
s mnm
i0
n
j0
For any double sequencex mn, we will define
The series
a mnis said to be summable|C, α, β| k,k ≥ 1, α, β > −1, if see 4
∞
m1
∞
n1
mn k−1Δ11σ m−1,n−1 αβ k
whereσ mn αβ denotes themn-term of the C, α, β transform of a sequence s mn, that is,
σ mn αβ 1
E α
m E β n
m
i0
n
j0
E α−1
Define
A2
k :
s mn∞
m,n0:
∞
m1
∞
n1
mn k−1 a mnk < ∞; a mn Δ11s m−1,n−1
8
fork ≥ 1.
A four-dimensional matrixT t mnij : m, n, i, j 0, 1, is said to be absolutely kth
power conservative, fork ≥ 1, if T ∈ BA2
k; that is, if
∞
m1
∞
n1
implies that
∞
m1
∞
n1
where
t mn∞
i0
∞
j0
t mnij s ij m, n 0, 1, . 11
Trang 3E Savas¸ et al 3
Theorem 1 C, α, β ∈ BA2
k for each α, β > −1.
Proof Let τ mn αβ denote themn-term of the C, α, β-transform, in terms of mna mn; that is,
τ mn αβ 1
E α
m E β n
m
i1
n
j1
E α−1
Forα, β > −1, since
τ mn αβ mnσ mn αβ − σ m,n−1 αβ − σ m−1,n αβ σ m−1,n−1 αβ , 13
to prove the theorem, it will be sufficient to show that
∞
m1
∞
n1
1
mn τ αβ
mnk
Using H ¨older’s inequality, we have
∞
m1
∞
n1
1
mn τ αβ
mnk
∞
m1
∞
n1
1
mn
1
E α
m E β n
m
i1
n
j1
E m−i α−1 E n−j β−1 ija ij
k
≤∞
m1
∞
n1
1
mnE α
m E β n
m
i1
n
j1
E m−i α−1 E β−1 n−j ij k a ijk×
1
E α
m E β n
m
i1
n
j1
E α−1 m−i E β−1 n−j
k−1
.
15 Since
1
E α
m E n β
m
i1
n
j1
E α−1 m−i E β−1 n−j 1, 16
we obtain
∞
m1
∞
n1
1
mn τ αβ
mnk
≤ ∞
m1
∞
n1
1
mnE α
m E β n
m
i1
n
j1
E α−1 m−i E β−1 n−j ij k a ijk
≤∞
i1
∞
j1
ij k a ijk∞
mi
∞
nj
E α−1 m−i E β−1 n−j mnE α
m E β n
.
17
Forα, β > −1 and m, n ≥ 1,
∞
mi
∞
nj
E α−1 m−i E n−j β−1 mnE α
m E n β
∞
mi
E α−1 m−i
mE α m
∞
nj
E β−1 n−j
nE β n
1
j
∞
mi
E α−1 m−i
mE α
m ij−1 18 Thus
∞
m1
∞
n1
1
mn τ αβ
mnk
O1∞
i1
∞
j1
ij k a ijk1
ij O1
∞
i1
∞
j1
ij k−1 a ijk O1 19
sinces mn ∈ A2
k
Trang 44 Journal of Inequalities and Applications
Using the notation of5,
θ α
E α m
m
i0
E α−1 m−i s in C, α, 0s mn
,
θ mn β : 1
E n β
n
j0
E β−1 n−j s mj C, 0, βs mn
,
σ mn: m 1n 11 m
i0
n
j0
s ij C, 1, 1s mn
20
Corollary 1 C, α, 0 ∈ BA2
k for each α > −1.
Corollary 2 C, 0, β ∈ BA2
k for each α > −1.
Corollary 3 C, 1, 1 ∈ BA2
k .
References
1 T M Flett, “On an extension of absolute summability and some theorems of Littlewood and Paley,”
Proceedings of the London Mathematical Society, vol 7, pp 113–141, 1957.
2 G Das, “A Tauberian theorem for absolute summability,” Mathematical Proceedings of the Cambridge
Philosophical Society, vol 67, pp 321–326, 1970.
3 E Savas¸ and H S¸evli, “On extension of a result of Flett for Ces´aro matrices,” Applied Mathematics Letters,
vol 20, no 4, pp 476–478, 2007.
4 B E Rhoades, “Absolute comparison theorems for double weighted mean and double Ces`aro means,”
Mathematica Slovaca, vol 48, no 3, pp 285–301, 1998.
5 M Y Mirza and B Thorpe, “Tauberian constants for double series,” Journal of the London Mathematical
Society, vol 57, no 1, pp 170–182, 1998.