  We consider the production of electron cyclotron maser emission by
low-density, highly magnetized plasmas in relativistic jets. The population
inversion required to drive cyclotron maser instability could occur in
localized, transient sites where hydromagnetic instabilities, shocks, and/or
turbulence lead to magnetic mirroring along current-carrying flux tubes. The
maser is pumped as electrons are accelerated by the parallel electric field
that develops as a result of the mirror. We estimate the maximum brightness
temperatures that can be obtained in a single maser site and in an array of
many masers operating simultaneously, under conditions likely to apply in
blazar jets. Synchrotron absorption, by relativistic electrons within the jet,
presents the largest obstacle to the escape of the maser radiation, and may
render most of it invisible. However, we argue that a high brightness
temperature could be produced in a thin boundary layer outside the synchrotron
photosphere, perhaps in the shear layer along the wall of the jet. Induced
Compton scattering provides additional constraints on the maximum brightness
temperature of a masing jet. We suggest that recent observations of diffractive
scintillation in the blazar J1819+3845, indicating intrinsic brightness
temperatures greater than 10^{14} K at 5 GHz, may be explained in terms of
cyclotron maser emission. High brightness temperature maser emission from
blazar jets may extend to frequencies as high as ~100 GHz, with the maximum
possible T_B scaling roughly as 1/frequency. Less massive relativistic jet
sources, such as microquasars, are even better candidates for producing
cyclotron maser emission, primarily in the infrared and optical bands.
