The quantum-classical partition-function inequality

Let $Z_q(\beta,h)=\sum_{n=1}^{\infty}e^{-\beta E_n(h)}$ be the quantum statistical sum, where $E_n(h)$ are the energy eigenvalues, and let

$$Z_c(\beta)=\int\int e^{-\beta H(p,q)}\,dp\,dq$$

be the classical statistical sum for the Hamiltonian

$$H(p,q)=\frac{1}{2m}\sum_{j=1}^{N}p_j^2+V(q).$$

Here $h>0$ is Planck's constant, $\beta>0$, and $N$ is the dimension of the coordinate space. Quantum-classical partition-function inequality. The inequality

$$(2\pi h)^N Z_q(\beta,h)\leq Z_c(\beta)$$

is true for all $h>0$ and $\beta>0$. This extends the Wigner--Kirkwood classical-limit relation from the limit $h\to0$ to a proposed global inequality, but the source supplies no resolution of the conjecture.