Global conservation constraints satisfied by the Vlasov-Maxwell equations are used to obtain a nonlinear bound on the unstable electrostatic fluctuation energy that can develop for non-relativistic non-neutral electron flow in a planar diode with an axial applied magnetic field. It is shown that the Heimholte free energy is a minimum for the thermal equilibrium reference state. The nonlinear bound on unstable fluctuation energy that can develop for general initial conditions is calculated for the case of flute perturbations with no axial dependence. To determine the lowest upper bound on fluctuation energy consistent with conservation constraints, the density, temperature and drift velocity of the reference state are chosen to minimize the nonlinear bound. The analysis assumes that the net flux of particles, momentum, and energy, vanish identically at the cathode and at the anode.