On cylindrically bounded $H$-Hypersurfaces of $\mathbb{H}^{n}\times   \mathbb{R}$
  We show that $H$-hypersurfaces of $\mathbb{H}^{n}\times \mathbb{R}$ contained in a vertical cylinder and with Ricci curvature with strong quadratic decay have mean curvature $| H| > (n-1)/n$. 
G. Pacelli Bessa, Silvana M. Costa