  The spectral bound, s(a A + b V), of a combination of a resolvent positive
linear operator A and an operator of multiplication V, was shown by Kato to be
convex in b \in R. This is shown here, through an elementary lemma, to imply
that s(a A + b V) is also convex in a > 0, and notably, \partial s(a A + b V) /
\partial a <= s(A) when it exists. Diffusions typically have s(A) <= 0, so that
for diffusions with spatially heterogeneous growth or decay rates, greater
mixing reduces growth. Models of the evolution of dispersal in particular have
found this result when A is a Laplacian or second-order elliptic operator, or a
nonlocal diffusion operator, implying selection for reduced dispersal. These
cases are shown here to be part of a single, broadly general, `reduction'
phenomenon.
