### On some convexity properties of generalized Cesàro sequence spaces.

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In this paper we define a generalized Cesàro sequence space $ces\left(p\right)$ and consider it equipped with the Luxemburg norm under which it is a Banach space, and we show that the space $ces\left(p\right)$ posses property (H) and property (G), and it is rotund, where $p=\left({p}_{k}\right)$ is a bounded sequence of positive real numbers with ${p}_{k}>1$ for all $k\in N$.

In this paper, we define the direct sum ${\left({\u2a01}_{i=1}^{n}{X}_{i}\right)}_{ce{s}_{p}}$ of Banach spaces X₁,X₂,..., and Xₙ and consider it equipped with the Cesàro p-norm when 1 ≤ p < ∞. We show that ${\left({\u2a01}_{i=1}^{n}{X}_{i}\right)}_{ce{s}_{p}}$ has the H-property if and only if each ${X}_{i}$ has the H-property, and ${\left({\u2a01}_{i=1}^{n}{X}_{i}\right)}_{ce{s}_{p}}$ has the Schur property if and only if each ${X}_{i}$ has the Schur property. Moreover, we also show that ${\left({\u2a01}_{i=1}^{n}{X}_{i}\right)}_{ce{s}_{p}}$ is rotund if and only if each ${X}_{i}$ is rotund.

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