Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 279421, 7 pages doi:10.1155/2009/279421 Research Article On Absolute Ces ` aro Summability Hamdullah S ¸ evli 1 and Ekrem Savas ¸ 2 1 Department of Mathematics, Faculty of Arts and Sciences, Y ¨ uz¨ unc ¨ uYıl University, 65080 Van, Turkey 2 Department of Mathematics, ˙ Istanbul Ticaret University, ¨ Usk ¨ udar 36472, ˙ Istanbul, Turkey Correspondence should be addressed to Hamdullah S ¸evli, hsevli@yahoo.com Received 14 July 2008; Accepted 7 June 2009 Recommended by L ´ aszl ´ o Losonczi Denote by A k the sequence space defined by A k {s n : ∑ ∞ n1 n k-1 |a n | k < ∞,a n s n - s n-1 } for k ≥ 1. In a recent paper by E. Savas ¸ and H. S ¸evli 2007, they proved every Ces` aro matrix of order α, for α> -1, C, α ∈ BA k for k ≥ 1. In this paper, we consider a further extension of absolute Ces` aro summability. Copyright q 2009 H. S ¸ evli and E. Savas ¸. 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. 1. Introduction Let ∑ a v denote a series with partial sums s n . For an infinite matrix T , t n , the nth term of the T -transform of s n is denoted by t n ∞ v0 t nv s v . 1.1 A series ∑ a v is said to be absolutely T -summable if ∑ n |Δt n-1 | < ∞, where Δ is the forward difference operator defined by Δt n-1 t n-1 - t n . Papers dealing with absolute summability date back at least as far as Fekete 1. A sequence s n is said to be of bounded variation bv if ∑ n |Δs n | < ∞. Thus, to say that a series is absolutely summable by a matrix T is equivalent to saying that the T -transform the sequence is in bv. Necessary and sufficient conditions for a matrix T :bv → bv are known. See, e.g., Stieglitz and Tietz 2. Let σ α n denote the nth terms of the transform of a Ces´ aro matrix C, α of a sequence s n . In 1957 Flett 3 made the following definition. A series ∑ a n , with partial sums s n , is