  The total mean curvature functional for submanifolds into the Riemannian
product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first
variational formula is presented. Later on, two second order differential
operators are defined and a nice integral inequality relating both of them is
proved. Finally we prove our main result: an integral inequality for closed
stationary $\mathcal{H}$-surfaces in $\mathbb{S}^n\times\mathbb{R}$,
characterizing the cases where the equality is attained.
