  We construct perturbation series for the q-th moment of eigenfunctions of
various critical random matrix ensembles in the strong multifractality regime
close to localization. Contrary to previous investigations, our results are
valid in the region q<1/2. Our findings allow to verify, at first leading
orders in the strong multifractality limit, the symmetry relation for anomalous
fractal dimensions Delta(q)=Delta(1-q), recently conjectured for critical
models where an analogue of the metal-insulator transition takes place. It is
known that this relation is verified at leading order in the weak
multifractality regime. Our results thus indicate that this symmetry holds in
both limits of small and large coupling constant. For general values of the
coupling constant we present careful numerical verifications of this symmetry
relation for different critical random matrix ensembles. We also present an
example of a system closely related to one of these critical ensembles, but
where the symmetry relation, at least numerically, is not fulfilled.
