  Let $H_1$ and $H_2$ be selfadjoint operators or relations (multivalued
operators) acting on a separable Hilbert space and assume that the inequality
$H_1 \leq H_2$ holds. Then the validity of the inequalities $-H_1^{-1} \leq
-H_2^{-1}$ and $H_2^{-1} \leq H_1^{-1}$ is characterized in terms of the
inertia of $H_1$ and $H_2$. Such results are known for matrices and boundedly
invertible operators. In the present paper those results are extended to
selfadjoint, in general unbounded, not necessarily boundedly invertible,
operators and, more generally, for selfadjoint relations in separable Hilbert
spaces.
