Slab theorem and halfspace theorem for constant mean curvature surfaces in $\mathbb H^2\times\mathbb R$

We prove that a properly embedded annular end of surface in $\mathbb{H}^2\times\mathbb{R}$ with constant mean curvature $0\<H\leq 1/2$ can not be contained any horizontal slab. Moreover, we show $\mathbb{H}^2 \times \[0,+\infty)$ and finite topology is necessarily graph over simply connected domain $\mathbb{H}^2$. For the case $H=1/2$, entire.