We consider the eigenvalue problem for the Schrödinger operator on bounded, convex domains with mixed boundary conditions, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. We prove inequalities between the lowest eigenvalues corresponding to two different choices of such boundary conditions on both planar and higher-dimensional domains. We also prove an inequality between higher order mixed eigenvalues and pure Dirichlet eigenvalues on multidimensional polyhedral domains.