  In a recent paper [J. Math. Phys. 47 082303 (2006)], Quantum Energy
Inequalities were used to place simple geometrical bounds on the energy
densities of quantum fields in Minkowskian spacetime regions. Here, we refine
this analysis for massive fields, obtaining more stringent bounds which decay
exponentially in the mass. At the technical level this involves the
determination of the asymptotic behaviour of the lowest eigenvalue of a family
of polyharmonic differential equations, a result which may be of independent
interest. We compare our resulting bounds with the known energy density of the
ground state on a cylinder spacetime. In addition, we generalise some of our
technical results to general $L^p$-spaces and draw comparisons with a similar
result in the literature.
