Mixed Hessian inequalities and uniqueness in the class $\mathcal{E}(X,\omega ,m)$

We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Amp`ere equation. Exploiting this and using Ko{\l}odziej’s mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact K"ahler manifolds in the case of probability measures vanishing on $m$-polar sets.