  A family of rigorous upper bounds on the growth rate of local gyrokinetic
instabilities in magnetized plasmas is derived from the evolution equation for
the Helmholtz free energy. These bounds hold for both electrostatic and
electromagnetic instabilities, regardless of the number of particle species,
their collision frequency, and the geometry of the magnetic field. A large
number of results that have earlier been derived in special cases and observed
in numerical simulations are thus brought into a unifying framework. These
bounds apply not only to linear instabilities but also imply an upper limit to
the nonlinear growth of the free energy.
