  We show that for any complete connected K\"ahler manifold the index of the
group of complex affine transformations in the group of c-projective
transformations is at most two unless the K\"ahler manifold is isometric to
complex projective space equipped with a positive constant multiple of the
Fubini-Study metric. This establishes a stronger version of the recently proved
Yano-Obata conjecture for complete K\"ahler manifolds.
