  We derive Lieb-Thirring inequalities for the Riesz means of eigenvalues of
order gamma >= 3/4 for fourth order Schr\"odinger operators in arbitrary
dimensions. We also consider some extensions to polyharmonic operators, and to
systems of such operators. For the critical case gamma = 1 - 1/2l in dimension
d=1 with differential order 2l >= 4 we prove the strict inequality
L^0(l,gamma,d) < L(l,gamma,d), which holds in contrast to current conjectures.
