uncommon questions - slow propagation of modes

Imagine a system which is described by the Lagrange function \(\mathcal{L} = \partial_\mu\Phi\partial^mu\Phi/2 - m^2\Phi^2/2\) and derive its equation of motion. Please derive the dispersion relation by substituting a plane wave \(\exp(\pm\mathrm{i}k_\sigma x^\sigma)\) and show that the group velocity \(\mathrm{d}\omega/\mathrm{d}k\) is smaller than \(c\) and that the phase velocity is larger than \(c\). Then, please show that the geometric average of phase- and group velocity is \(c\).