  The Coma cluster exhibits evidence for a high-energetic non-thermal particle
population. At frequencies >1 GHz recent radio halo observations confirm a
significant spectral steepening of the volume-integrated emission. We calculate
the volume-averaged high-energy spectrum due to inverse Compton scattering off
the CMB radiation field and non-thermal bremsstrahlung according to an
exponential cutoff in the electron spectrum as deduced from the radio
observations. The synchrotron radiation from secondary pairs, created from the
decay of charged mesons produced in hadronic pp-interactions, is found to set
significant constraints on the energy content of relativistic hadrons in Coma.
This limits the maximum flux at high energies. Our findings support a low ratio
of relativistic hadron to thermal energy density. Predictions for Coma's high
energy emission are discussed in the light of current and expected abilities of
upcoming gamma-ray instruments.
