  The eigenvalue problem for the Laplacian on bounded, planar, convex domains
with mixed boundary conditions is considered, where a Dirichlet boundary
condition is imposed on a part of the boundary and a Neumann boundary condition
on its complement. Given two different such choices of boundary conditions for
the same domain, we prove inequalities between their lowest eigenvalues. As a
special case, we prove parts of a conjecture on the order of mixed eigenvalues
of triangles.
