  This work introduces a limitation on the minimum value that can be assumed by
the energy of a relativistic gas in the presence of a non-zero heat flux. Such
a limitation arises from the non-negativity of the particle distribution
function, and is found by solving the Hamburger moment problem. The resulting
limitation is seen to recover the Taub inequality in the case of a zero heat
flux, but is more strict if a non-zero heat flux is considered. These results
imply that, in order for the distribution function to be non-negative, (i) the
energy of a gas must be larger than a minimum threshold; (ii) the heat flux, on
the other hand, has a maximum value determined by the energy and the pressure
tensor; and (iii) there exists an upper limit for the the adiabatic index
$\Gamma$ of the relativistic equation of state, and that limit decreases in the
presence of a heat flux and pressure anisotropy, asymptoting to a value $\Gamma
= 1$. The latter point implies that the Synge equation of state is formally
incompatible with a relativistic gas showing a heat flux, except in certain gas
states.
