  We investigate how the presence of a non-thermal tail beyond a Maxwellian
electron distribution affects the synchrotron process as well as Comptonization
in plasmas with parameters typical for accretion flows onto black holes. We
find that the presence of the tail can significantly increase the net (after
accounting for self-absorption) cyclo-synchrotron emission of the plasma, which
emission then provides seed photons for Compton upscattering. Thus, the
luminosity in the thermally-Comptonized spectrum is enhanced as well. The
importance of these effects increases with both increasing Eddington ratio and
the black hole mass. The enhancement of the Comptonized synchrotron luminosity
can be as large as by factors of $\sim 10^3$ and $\sim 10^5$ for stellar and
supermassive black holes, respectively, when the energy content in the
non-thermal tail is 1 per cent.
  The presence of the tail only weakly hardens the thermal Comptonization
spectrum but it leads to formation of a high-energy tail beyond the thermal
cut-off, which two effects are independent of the nature of the seed photons.
Since observations of high-energy tails in Comptonization spectra can constrain
the non-thermal tails in the electron distribution and thus the Comptonized
synchrotron luminosity, they provide upper limits on the strength of magnetic
fields in accretion flows. In particular, the measurement of an MeV tail in the
hard state of Cyg X-1 by McConnell et al. implies the magnetic field strength
in this source to be at most an order of magnitude below equipartition.
