Conditions for a ferromagnetic ground state of Heisenberg Hamiltonians

For isotropic Heisenberg Hamiltonians with ferromagnetic nearest and anitferromagnetic next-nearest neighbor interactions, $H=J_1\Sigma \stackrel{}{\left(\mathrm{nn}\right)}{\overrightarrow{\mathrm{S}}}_n·{\overrightarrow{\mathrm{S}}}_{n+{\delta }_1}+J_2\Sigma \stackrel{}{\left(\mathrm{nnn}\right)}{\overrightarrow{\mathrm{S}}}_n·{\overrightarrow{\mathrm{S}}}_{n+{\delta }_2},$ $J_1<0, J_2>0\mathrm{}$ the following conditions for a ferromagnetic ground state are derived. With periodic boundary conditions it is necessary that we have $\frac{\left|J_1\right|}{J_2}>4\mathrm{}{cos}^2\left(\frac{\pi }{N}\right)$,2,4 for the simple-cubic lattice in one, two, and three dimensions, respectively. For periodic and for open boundaries it is sufficient that we have $\frac{\left|J_1\right|}{J_2}>4.2+\frac{1}{s}$, and $4+\frac{2}{s}$ in the three cases.