  We provide complementary semiclassical bounds for the Riesz means $R_1(z)$ of
the eigenvalues of various biharmonic operators, with a second term in the
expected power of $z$. The method we discuss makes use of the averaged
variational principle (AVP), and yields two-sided bounds for individual
eigenvalues, which are semiclassically sharp. The AVP also yields comparisons
with Riesz means of different operators, in particular Laplacians.
