Threshold inequalities for ferromagnetic ground state of Ising and Heisenberg models

For the Ising, the classical, and the quantum-mechanical Heisenberg models with ferromagnetic nearest- and antiferromagnetic next-nearest-neighbor interactions the inequalities ${\alpha }^{\left(\mathrm{ls}\right)}<~{\alpha }_{\mathrm{cl}}^{\left(\mathrm{Heis}\right)}<~{\alpha }_{\mathrm{qm}}^{\left(\mathrm{Heis}\right)}$ and ${\alpha }^{\left(\mathrm{ls}\right)}<{\alpha }_{\mathrm{qm}}^{\left(\mathrm{Heis}\right)}$ are proved for the threshold of the coupling-constant ratio $\frac{J_1}{J_2}=\alpha$ for the ferromagnetic ground state. Combined with the classical spin limit they predict a spin dependence of ${\alpha }_{\mathrm{qm}}^{\left(\mathrm{Heis}\right)}$. This prediction remains valid for systems with $n\mathrm{th}$-nearest-neighbor interactions.