  Using a variational method, we derive an exact upper bound for one-magnon
excitation energy in ferromagnetic spinor gases, which limits the quantum
corrections to the effective mass of a magnon to be positive. We also derive an
upper bound for one-magnon excitation energy in lattice systems. The results
hold for both Bose and Fermi systems in $d$ dimensions as long as the
interaction is local and invariant under spin rotation.
