Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 86757, 6 pages doi:10.1155/2007/86757 Research Article Some Geometric Inequalities in a New Banach Sequence Space M. Mursaleen, Rifat C¸ olak, and Mikail Et Received 11 July 2007; Accepted 18 November 2007 Recommended by Peter Yu Hin Pang The difference sequence space m(φ, p,Δ (r) ), which is a gener alization of the s pace m(φ) introduced and studied by Sargent (1960), was defined by C¸ olak and Et (2005). In this paper we establish some geometric inequalities for this space. Copyright © 2007 M. Mursaleen et al. This is an op en 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 and preliminaries Let Ꮿ denote the space whose elements are finite sets of distinct positive integers. Given an element σ ∈ Ꮿ,wewritec(σ) for the sequence (c n (σ)) such that c n (σ) = 1forn ∈ σ, and c n (σ) = 0, otherwise. Further Ꮿ s =  σ ∈ Ꮿ : ∞  n=1 c n (σ) ≤ s  , (1.1) that is, Ꮿ s is the set of those σ whose support has cardinality at most s,wheres is a natural number. Let w be the set of all real sequences and Φ =  φ =  φ n  ∈ w : φ 1 > 0,∇φ k ≥ 0, ∇  φ k k  ≤ 0(k = 1,2, )  , (1.2) where ∇φ k = φ k − φ k−1 . For φ ∈ Φ,Sargent[1] introduced the following sequence space: m(φ) =  x =  x n  ∈ w :sup s≥1 sup σ∈Ꮿ s  1 φ s  n∈σ |x n |  < ∞  . (1.3) 2 Journal of Inequalities and Applications In [2], the space m(φ) has been considered for matrix transformations and in [3]some of its geometric properties have been considered. Tripathy and Sen [4]extendedm(φ)to m(φ, p),1 ≤ p<∞. Recently, C¸olakandEt[5] defined the space m(φ, p,Δ (r) ) by using the idea of difference sequences (see [6–8]). Let r be a positive integer throughout. The operators Δ (r) ,Σ (r) : w→w are defined by  Δ (1) x  k = (Δx) k = x k − x k+1 ,  Σ (1) x  k = (Σx) k = ∞  j=k x j (k = 1,2, ), Δ (r) = Δ (1) ◦ Δ (r−1) , Σ (r) = Σ (1) ◦ Σ (r−1) ,(r ≥ 2), Σ (r) ◦ Δ (r) = Δ (r) ◦ Σ (r) = id, the identity on w. (1.4) For 0 ≤ p<∞, the space m(φ, p,Δ (r) )isdefinedasfollows: m  φ, p,Δ (r)  =  x ∈ w :sup s≥1,σ∈Ꮿ s  1 φ s  n∈σ   Δ (r) x n   p  < ∞  , (1.5) which is a Banach space (1 ≤ p<∞) with the norm x m(φ,p,Δ (r) ) = r  i=1 |x i | +sup s≥1,σ∈Ꮿ s 1 φ s   n∈σ   Δ (r) x n   p  1/p , (1.6) and a complete p-normed space (0 <p<1) with the p-norm x m p (φ,Δ (r) ) = r  i=1 |x i | p +sup s≥1,σ∈Ꮿ s 1 φ s  n∈σ   Δ (r) x n   p . (1.7) In this paper, we will consider the case 1 <p< ∞ to study some geometric properties of m(φ, p,Δ (r) ). We will examine the Banach-Saks property of type p,strictconvexityand uniform convexity. The space m(φ, p),1 ≤ p<∞ was defined by Tripathy and Sen [4] which is in fact m(φ, p,Δ)withΔ replaced by id. Let 1 <p< ∞. ABanachspaceX is said to have the Banach-Saks property of type p or property (BS) p ifeveryweaklynull-sequence(x k ) has a subsequence (x k i )suchthatfor some C>0, the inequality      n  i=0 x k i      X ≤ c(n +1) 1/p , n = 1,2,3, , (1.8) holds. The property (BS) p for a Ces ` aro sequence space was considered in [9]. We find uniform convexity and strict convexity of our space through the Gurarii’s modulus of convexity (see [10, 11]). For a normed linear space X, the modulus of convexity defined by β X (ε) = inf  1 − inf 0≤α≤1 αx +(1− α)y : x, y ∈ S(X),x − y=ε  , (1.9) M. Mursaleen et al. 3 is called the Gurarii’s modulus of convexity, where S(X) denotes the unit sphere in X and 0 <ε ≤ 2. If 0 <β X (ε) < 1, then X is uniformly convex and if β X (ε) ≤ 1, then X is strictly convex. 2. Main results Theorem 2.1. The space m(φ, p,Δ (r) ) has the Banach-Saks property of type p. Proof. We will prove the case r = 1 and the general case can be followed on the same lines.  Let (ε n ) be a sequence of positive numbers for which  ∞ n=1 ε n ≤ 1/2. Let (x n )bea weakly null sequence in B(m(φ, p,Δ)), the unit ball in m(φ, p,Δ). Set x 0 = 0andz 1 = x n 1 = Δx 1 . Then there exists s 1 ∈ N such that       i∈τ 1 z 1 (i)e i      m(φ,p,Δ) <ε 1 , (2.1) where τ 1 consists of the elements of σ which exceed s 1 . Since x n w −→ 0 ⇒ x n →0coordinate- wise, there is n 2 ∈ N such that      s 1  i=1 x n (i)e i      m(φ,p,Δ) <ε 1 ,whenn ≥ n 2 . (2.2) Set z 2 = x n 2 = Δx 2 . Then there exists s 2 >s 1 such that       i∈τ 2 z 2 (i)e i      m(φ,p,Δ) <ε 2 , (2.3) where τ 2 consists of the elements of σ which exceed s 2 . Again using the fact x n →0coordi- natewise, there exists n 3 >n 2 such that      s 2  i=1 x n (i)e i      m(φ,p,Δ) <ε 2 ,whenn ≥ n 3 . (2.4) Continuing this process, we can find two increasing sequences (s i )and(n i )suchthat      s j  i=1 x n (i)e i      m(φ,p,Δ) <ε j ,whenn ≥ n j+1 ,       i∈τ j z j (i)e i      m(φ,p,Δ) <ε j , (2.5) where z j = x n j = Δx j and τ j consists of the elements of σ which exceed s j . Note that z j (i) is a term in the sequence with fixed j and running i. 4 Journal of Inequalities and Applications Since ε j−1 + ε j < 1, we have  1 φ s  n∈σ   z j (n)    ≤  ε j−1 + ε j  < 1, (2.6) for all j ∈ N and s ≥ 1. Hence      n  j=1 z j      m(φ,p,Δ) =      n  j=1  s j−1  i=1 z j (i)e i + s j  i=s j−1 +1 z j (i)e i +  i∈τ j z j (i)e i       m(φ,p,Δ) ≤      n  j=1  s j−1  i=1 z j (i)e i       m(φ,p,Δ) +      n  j=1  s j  i=s j−1 +1 z j (i)e i       m(φ,p,Δ) +      n  j=1   i∈τ j z j (i)e i       m(φ,p,Δ) ≤ n  j=1       s j  i=s j−1 +1 z j (i)e i       m(φ,p,Δ) +2 n  j=1 ε j , n  j=1      s j  i=s j−1 +1 z j (i)e i      p m(φ,p,Δ) = n  j=1 sup s≥1 sup τ j−1 ∈Ꮿ s  1 φ s  i∈τ j−1   z j (i)   p  ≤ n  j=1 sup s≥1 sup σ∈Ꮿ s  1 φ s  i∈σ   z j (i)   p  ≤ n. (2.7) Therefore by (2.7)      n  j=1 z j      m(φ,p,Δ) ≤ n 1/p +1≤ 2n 1/p (2.8) since  n j =1 ε j ≤ 1/2. Hence m(φ, p,Δ) has the Banach-Saks property of type p. Remark 2.2. The above result can also be extended to the case when r =1 and so the proof should also work for a more general case with Δ replaced by a matrix operator (transformation). Theorem 2.3. The Gurarii’s modulus of convexity for the space X = m(φ, p,Δ) is β X (ε) ≤ 1 −  1 −  ε 2  p  1/p , (2.9) where 0 <ε ≤ 2. M. Mursaleen et al. 5 Proof. Let x ∈ m(φ, p,Δ). Then x m(φ,p,Δ) =Δx m(φ,p) =   x 1   +sup s≥1,σ∈Ꮿ s 1 φ s   n∈σ   Δx n   p  1/p . (2.10) Let 0 <ε ≤ 2 and consider the sequences u = (u n ) =    1 −  ε 2  p  1/p ,   ε 2  ,0,0,  , v = (v n ) =    1 −  ε 2  p  1/p ,   − ε 2  ,0,0,  . (2.11) Then Δu m(φ,p) =u m(φ,p,Δ) =1,Δv m(φ,p) =v m(φ,p,Δ) =1, that is, u,v ∈ S(m(φ, p, Δ)) and Δu −Δv m(φ,p) =u − v m(φ,p,Δ) = ε. For 0 ≤ α ≤ 1,   αu +(1− α)v   p m(φ,p,Δ) =   αΔu +(1− α)Δv   p m(φ,p) = 1 −  ε 2  p + |2α − 1|  ε 2  p . (2.12) Hence inf 0≤α≤1   αu +(1− α)v   p m(φ,p,Δ) = 1 −  ε 2  p . (2.13) Therefore, for p ≥ 1 β X (ε) ≤ 1 −  1 −  ε 2  p  1/p . (2.14) This completes the proof of the theorem.  Corollary 2.4. (i) If ε = 2, then β X (ε) ≤ 1 and hence m(φ, p,Δ) is strictly convex. (ii) If 0 <ε<2, then 0 <β X (ε) < 1 and hence m(φ, p,Δ) is uniformly convex. Remark 2.5. Note that these results are best possible for the time being, that is, they cannot be readily generalized to the gener al case because our results also hold for general matrix transformation. Acknowledgments The present paper was completed when Professor Mursaleen v isited Firat University (May-June, 2007). The author is very much grateful to the Firat University for provid- ing hospitalities. This research was supported by FUBAP (The Management Union of the Scientific Research Projects of Firat University) when the first author visited Firat Univer- sity under the Project no. 1179. 6 Journal of Inequalities and Applications References [1] W.L.C.Sargent,“Somesequencespacesrelatedtothel p spaces,” Journal of the London Mathe- matical Society, vol. 35, no. 2, pp. 161–171, 1960. [2] E. Malkowsky and M. Mursaleen, “Matrix transformations between FK-spaces and the sequence spaces m(φ)andn(φ),” Journal of Mathematical Analysis and Applications, vol. 196, no. 2, pp. 659–665, 1995. [3] M. Mursaleen, “Some geometric properties of a sequence space related to l p ,” Bulletin of the Australian Mathematical Soc iety, vol. 67, no. 2, pp. 343–347, 2003. [4] B. C. Tripathy and M. Sen, “On a new class of sequences related to the space l p ,” Tamkang Journal of Mathematics, vol. 33, no. 2, pp. 167–171, 2002. [5] R. C¸ olak and M. 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Ull ´ an, “Some properties of Gurarii’s modulus of convexity,” Archiv der Math- ematik, vol. 71, no. 5, pp. 399–406, 1998. M. Mursaleen: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Email address: mursaleenm@gmail.com Rifat C¸ olak: Department of Mathematics, Firat University, 23119 Elazı ˘ g, Turkey Email address: rcolak@firat.edu.tr Mikail Et: Department of Mathematics, Firat University, 23119 Elazı ˘ g, Turkey Email address: mikailet@yahoo.com . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 86757, 6 pages doi:10.1155/2007/86757 Research Article Some Geometric Inequalities in a New Banach. order m and matrix transformations,” Acta Mathematica Sinica, vol. 23, no. 3, pp. 521–532, 2007. [9] Y. Cui and H. Hudzik, “On the Banach- Saks and weak Banach- Saks properties of some Banach sequence. Mursaleen: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India Email address: mursaleenm@gmail.com Rifat C¸ olak: Department of Mathematics, Firat University, 23119 Elazı ˘ g,