  This paper considers Lieb-Thirring inequalities for higher order differential
operators. A result for general fourth-order operators on the half-line is
developed, and the trace inequality tr((-Delta)^2 - C^{HR}_{d,2} / (|x|^4) -
V(x))^{-\gamma} < C_\gamma \int_{R^d} V(x)_+^{\gamma + d/4} dx for gamma \geq 1
- d/4, where C^{HR}_{d,2} is the sharp constant in the Hardy-Rellich inequality
and where C_\gamma > 0 is independent of V, is proved for dimensions d = 1,3.
As a corollary of this inequality a Sobolev-type inequality is obtained.
