  The H.E.S.S. (High Energy Stereoscopic System) telescopes observed Coma for
~8hr in a search for gamma-ray emission at energies >1TeV. The large 3.5deg
FWHM field of view of H.E.S.S. is ideal for viewing a range of targets at
various sizes including the Coma cluster core, the radio-relic (1253+275) and
merger/infall (NGC 4839) regions to the southwest, and features greater than
deg away. No evidence for point-like nor extended TeV gamma-ray emission was
found and upper limits to the TeV flux F(E) for E>1, >5, and >10TeV were set
for the Coma core and other regions. Converting these limits to an energy flux
E^2F(E) the lowest or most constraining is the E>5TeV upper limit for the Coma
core (0.2deg radius) at ~8Crab flux units or ~10^{-13}ph cm^{-2} s^{-1}. The
upper limits for the Coma core were compared with a prediction for the
gamma-ray emission from proton--proton interactions, the level of which
ultimately scales with the mass of the Coma cluster. A direct constraint using
our most stringent limit for E>5 TeV, on the total energy content in
non-thermal protons with injection energy spectrum proportional to E^{-2.1} and
spatial distribution following the thermal gas in the cluster, is found to be
~0.2 times the thermal energy, or ~10^{62}erg. The E>5 TeV gamma-ray threshold
in this case corresponds to cosmic-ray proton energies >50TeV. Our upper limits
rule out the most optimistic theoretical models for gamma ray emission from
clusters and complement radio observations which constrain the cosmic ray
content in clusters at significantly lower proton energies, subject to
assumptions on the magnetic field strength.
