© Hindawi Publishing Corp. ON k-NEARLY UNIFORM CONVEX PROPERTY IN GENERALIZED CESÀRO SEQUENCE SPACES

Wedefine a generalized Cesàro sequence space ces(p), wherep=(pk) is a bounded sequence of positive real numbers, and consider it equipped with the Luxemburg norm. The main purpose of this paper is to show that ces(p) is k-nearly uniform convex (k-NUC) for k ≥ 2 when limn→ ∞ infpn> 1. Moreover, we also obtain that the Cesàro sequence space cesp (where 1 < p <∞) is k-NUC, kR, NUC, and has a drop property. 2000 Mathematics Subject Classification: 46B20, 46B45. 1. Introduction. Let (X,‖·‖) be a real Banach space and let B(X) and S(X) be the closed unit ball and the unit sphere of X, respectively. For any sub-set A of X, we denote by conv(A) (resp., conv(A)) the convex hull (resp., the closed convex hull) of Clarkson [1] who introduced the concept of uniform convexity, and it is known that uniform convexity implies reflexivity of Banac