  We consider the magnetic Laplacian with the homogeneous magnetic field in two
and three dimensions. We prove that the $(k+1)$-th magnetic Neumann eigenvalue
of a bounded convex planar domain is not larger than its $k$-th magnetic
Dirichlet eigenvalue. In three dimensions, we restrict our attention to convex
domains, which are invariant under rotation by an angle of $\pi$ around an axis
parallel to the magnetic field. For such domains, we prove that the $(k+2)$-th
magnetic Neumann eigenvalue is not larger than the $k$-th magnetic Dirichlet
eigenvalue provided that this Dirichlet eigenvalue is simple. The proofs rely
on a modification of the strategy due to Levine and Weinberger.
