  Upper bounds on the growth of instabilities in gyrokinetic systems have
recently been derived by considering the optimal perturbations that maximise
the growth of a chosen energy norm. This technique has previously been applied
to two-species gyrokinetic systems with fully kinetic ions and electrons.
However, in tokamaks and stellarators, the expectation from linear instability
analyses is that the most important kinetic-electron contribution to ion-scale
modes comes from the trapped electrons, which bounce faster than the timescale
upon which instabilities evolve. As a result, a fully-kinetic electron response
is not required to describe unstable modes in most cases. Here, we apply the
optimal mode analysis to a reduced two-species system that consists of fully
gyrokinetic ions and bounce-averaged electrons with the aim of finding a
tighter bound on ion-scale instabilities in toroidal geometry. This analysis
yields bounds that are greatly reduced in comparison to the earlier two-species
result. Moreover, if the energy norm is properly chosen, wave-particle
resonance effects can be captured, reproducing the stabilisation of
density-gradient-driven instabilities in maximum-$J$ devices. The optimal mode
analysis also reveals that the maximum-$J$ property has an additional
stabilising effect on ion-temperature-gradient-driven instabilities, even in
the absence of an electron-free energy source. This effect is explained in
terms of the concept of mode inertia, making it distinct from other mechanisms.
