  We establish a lower bound for the eigenvalues of the Dirac operator defined
on a compact K\"ahler-Einstein manifold of positive scalar curvature and
endowed with particular ${\rm spin}^c$ structures. The limiting case is
characterized by the existence of K\"ahlerian Killing ${\rm spin}^c$ spinors in
a certain subbundle of the spinor bundle. Moreover, we show that the Clifford
multiplication between an effective harmonic form and a K\"ahlerian Killing
${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case
the result of A. Moroianu stating that, on a compact K\"ahler-Einstein manifold
of complex dimension $4\ell+3$ carrying a complex contact structure, the
Clifford multiplication between an effective harmonic form and a K\"ahlerian
Killing spinor is zero.
