Hindawi 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¸andS¸evli 2007, 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 k be a series with partial sums s n , A an infinite matrix. Then  a k is said to be absolutely summable A of order k ≥ 1if ∞  n1 n k−1   T n−1 − T n   k < ∞, 1 where T n : ∞  k0 a nk s k . 2 Denote by A k the sequence space defined by A k    s n  : ∞  n1 n k−1   a n   k < ∞; 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 ∈ BA k ,if T : A k →A k . In 1970, Das 2 defined such a matrix to be absolutely kth power conservative 2 Journal of Inequalities and Applications for k ≥ 1. In that paper, he proved that every conservative Hausdorff matrix H ∈ BA 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 ∈ BA k  for k ≥ 1. Since the Ces ´ aro matrices of order α for −1 <α<0arenot conservative, their result shows that being conservative is not a necessary condition for being absolutely kth power conservative. In this paper, we extend the result of 3 to double summability, thereby demonstrating that the property of being conservative is again not necessary for doubly infinite matrices to be absolutely kth power conservative. Let  ∞ m0  ∞ n0 a mn be an infinite double series with real or complex numbers, with partial sums s mn  m  i0 n  j0 a ij . 4 For any double sequence x mn , we will define Δ 11 x mn  x mn − x m1,n − x m,n1  x m1,n1 . 5 The series  a mn is said to be summable |C, α, β| k , k ≥ 1, α, β > −1, if see 4 ∞  m1 ∞  n1 mn k−1   Δ 11 σ αβ m−1,n−1   k < ∞, 6 where σ αβ mn denotes the mn-term of the C, α, β transform of a sequence s mn ,thatis, σ αβ mn  1 E α m E β n m  i0 n  j0 E α−1 m−i E β−1 n−j s ij . 7 Define A 2 k :   s mn  ∞ m,n0 : ∞  m1 ∞  n1 mn k−1   a mn   k < ∞; a mn Δ 11 s m−1,n−1  8 for k ≥ 1. A four-dimensional matrix T t mnij : m, n, i,j  0, 1,  is said to be absolutely kth power conservative, for k ≥ 1, if T ∈ BA 2 k ;thatis,if ∞  m1 ∞  n1 mn k−1   Δ 11 s m−1,n−1   k < ∞ 9 implies that ∞  m1 ∞  n1 mn k−1   Δ 11 t m−1,n−1   k < ∞, 10 where t mn  ∞  i0 ∞  j0 t mnij s ij m, n  0, 1, . 11 E. Savas¸etal. 3 Theorem 1. C, α, β ∈ BA 2 k  for each α, β > −1. Proof. Let τ αβ mn denote the mn-term of the C, α, β-transform, in terms of mna mn ;thatis, τ αβ mn  1 E α m E β n m  i1 n  j1 E α−1 m−i E β−1 n−j ija ij . 12 For α, β > −1, since τ αβ mn  mn  σ αβ mn − σ αβ m,n−1 − σ αβ m−1,n  σ αβ m−1,n−1  , 13 to prove the theorem, it will be sufficient to show that ∞  m1 ∞  n1 1 mn   τ αβ mn   k < ∞. 14 Using H ¨ older’s inequality, we have ∞  m1 ∞  n1 1 mn   τ αβ mn   k  ∞  m1 ∞  n1 1 mn      1 E α m E β n m  i1 n  j1 E α−1 m−i E β−1 n−j ija ij      k ≤ ∞  m1 ∞  n1 1 mnE α m E β n m  i1 n  j1 E α−1 m−i E β−1 n−j ij k   a ij   k ×  1 E α m E β n m  i1 n  j1 E α−1 m−i E β−1 n−j  k−1 . 15 Since 1 E α m E β n m  i1 n  j1 E α−1 m−i E β−1 n−j  1, 16 we obtain ∞  m1 ∞  n1 1 mn   τ αβ mn   k ≤ ∞  m1 ∞  n1 1 mnE α m E β n m  i1 n  j1 E α−1 m−i E β−1 n−j ij k   a ij   k ≤ ∞  i1 ∞  j1 ij k   a ij   k ∞  mi ∞  nj E α−1 m−i E β−1 n−j mnE α m E β n . 17 For α, β > −1andm, n ≥ 1, ∞  mi ∞  nj E α−1 m−i E β−1 n−j mnE α m E β n  ∞  mi E α−1 m−i mE α m ∞  nj E β−1 n−j nE β n  1 j ∞  mi E α−1 m−i mE α m ij −1 . 18 Thus ∞  m1 ∞  n1 1 mn   τ αβ mn   k  O1 ∞  i1 ∞  j1 ij k   a ij   k 1 ij  O1 ∞  i1 ∞  j1 ij k−1   a ij   k  O1 19 since s mn  ∈A 2 k . 4 Journal of Inequalities and Applications Using the notation of 5, θ α mn : 1 E α m m  i0 E α−1 m−i s in C, α, 0  s mn  , θ β mn : 1 E β n n  j0 E β−1 n−j s mj C, 0,β  s mn  , σ mn : 1 m  1n  1 m  i0 n  j0 s ij C, 1, 1  s mn  . 20 Corollary 1. C, α, 0 ∈ BA 2 k  for each α>−1. Corollary 2. C, 0,β ∈ BA 2 k  for each α>−1. Corollary 3. C, 1, 1 ∈ BA 2 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. . 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 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. extend the result of 3 to double summability, thereby demonstrating that the property of being conservative is again not necessary for doubly infinite matrices to be absolutely kth power conservative. Let  ∞ m0  ∞ n0 a mn be