The quantum-classical mean-energy inequality

Let $Z_q(\beta,h)=\sum_{n=1}^{\infty}e^{-\beta E_n(h)}$ and $Z_c(\beta)=\int\int e^{-\beta H(p,q)}\,dp\,dq$ be the quantum and classical statistical sums for the same system, with Hamiltonian $H(p,q)=\frac{1}{2m}\sum_{j=1}^{N}p_j^2+V(q)$. Define the quantum and classical mean energies by

$$E_q(\beta,h)=\frac{\sum_{n=1}^{\infty}E_n(h)e^{-\beta E_n(h)}}{Z_q(\beta,h)},\qquad E_c(\beta)=\frac{\int\int H(p,q)e^{-\beta H(p,q)}\,dp\,dq}{Z_c(\beta)}.$$

Quantum-classical mean-energy inequality. The inequality

$$E_q(\beta,h)\geq E_c(\beta)$$

is true for all $h>0$ and $\beta>0$. The claim proposes that the quantum mean energy always dominates the corresponding classical mean energy, but the source supplies no resolution.