  We derive explicit expressions for the sum rules of the eigenvalues of
inhomogeneous strings with arbitrary density and with different boundary
conditions. We show that the sum rule of order $N$ may be obtained in terms of
a diagrammatic expansion, with $(N-1)!/2$ independent diagrams. These sum rules
are used to derive upper and lower bounds to the energy of the fundamental mode
of an inhomogeneous string; we also show that it is possible to improve these
approximations taking into account the asymptotic behaviour of the spectrum and
applying the Shanks transformation to the sequence of approximations obtained
to the different orders. We discuss three applications of these results.
