  We investigate the effect of electronic correlations on the coupling of
electrons to Holstein phonons in the one-band Hubbard model. We calculate the
static electron-phonon vertex within linear response of Kotliar-Ruckenstein
slave-bosons in the paramagnetic saddle-point approximation. Within this
approach the on-site Coulomb interaction U strongly suppresses the coupling to
Holstein phonons at low temperatures. Moreover the vertex function does not
show particularly strong forward scattering. Going to larger temperatures
kT\sim t we find that after an initial decrease with U, the electron-phonon
coupling starts to increase with U, confirming a recent result of Cerruti,
Cappelluti, and Pietronero. We show that this behavior is related to an unusual
reentrant behavior from a phase separated to a paramagnetic state upon
decreasing the temperature.
