  We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and
Neumann boundary conditions. For a certain class of bounded, simply connected
planar domains we prove monotonicity properties of the first eigenfunction. As
a consequence, we establish a variant of the hot spots conjecture for mixed
boundary conditions. Moreover, we obtain an inequality between the lowest
eigenvalue of this mixed problem and the lowest eigenvalue of the corresponding
dual problem where the Dirichlet and Neumann boundary conditions are
interchanged. The proofs are based on a novel variational principle, which we
establish.
