  Within the time-dependent Gutzwiller approximation (TDGA) applied to
Holstein- and SSH-Hubbard models we study the influence of electron
correlations on the phonon self-energy. For the local Holstein coupling we find
that the phonon frequency renormalization gets weakened upon increasing the
onsite interaction $U$ for all momenta. In contrast, correlations can enhance
the phonon frequency shift for small wave-vectors in the SSH-Hubbard model.
Moreover the TDGA applied to the latter model provides a mechanism which leads
to phonon frequency corrections at intermediate momenta due to the coupling
with double occupancy fluctuations. Both models display a shift of the
nesting-induced to a $q=0$ instability when the onsite interaction becomes
sufficiently strong and thus establishing phase separation as a generic
phenomenon of strongly correlated electron-phonon coupled systems.
