On generalized quasi-convex bounded sequences

The space of all sequences a = (a(k)) for which parallel to a parallel to(q) = Sigma(k)k vertical bar Delta(2) a(k)vertical bar + sup(k) vertical bar a(k)vertical bar < infinity is denoted by q. Here, Delta a(k) = a(k) - a(k+1) and Delta(m)a(k) = Delta(Delta(m-1) a(k)) = Delta(m-1) a(k) - Delta(m-1) a(k+1) with Delta(0)a(k) = a(k), m >= 1. If a = (a(k)) is an element of q then k Delta a(k) -> 0 (k -> infinity) and q subset of by, the space of all sequences of bounded-variation, since Sigma vertical bar Delta a(k)vertical bar <= Sigma(k)k vertical bar Delta(2) a(k)vertical bar. In this study, we give a generalization of quasi-convex bounded sequences.