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The Hardy space $H_{\lambda}^{p}(\RR^{2}_+)$ associated with the Dunkl transform $\SF_{\lambda}$ and the Dunkl operator $D_x$\,(\,where$D_xf(x)=f'(x)+\frac{\lambda}{x}[f(x)-f(-x)]$\,) is the set of functions $F(x, y)=u(x, y)+iv(x, y)$ on the upper half plane $\RR_+^2=\{(x, y): x\in\RR, y>0\}$ satisfying the $\lambda$-Cauchy-Riemann equations: $D_xu-\partial_y v=0, \partial_y u +D_xv=0$ and $\sup_{y>0}\int_{\RR}|F(x, y)||x|^{2\lambda}dx<0$. In this paper, we will prove that the Ces\`{a}ro operator is bounded in the Dunkl Hardy space $H_{\lambda}^{p}(\RR^{2}_+)$ for every $p\geq\frac{2\lambda}{2\lambda+1},\,\lambda>0$:

where $C$ is a constant depending on $\alpha$, $p$, and $\lambda$. The average function for the Ces\`{a}ro operator $C_{\alpha}$ is $\phi_{\alpha}(t)=\alpha(1-t)^{\alpha-1}$ with $\alpha>0$.

2000 MS Classification: