  Upper bounds on the growth of free energy in gyrokinetics are derived. These
bounds apply to all local gyrokinetic instabilities in the geometry of a flux
tube, i.e. a slender volume of plasma aligned with the magnetic field,
regardless of the geometry of field, the number of particle species, or
collisions. The results apply both to linear instabilities and to the nonlinear
growth of finite-amplitude fluctuations.
