In this article we study the behaviour of dominant Fredholm eigenvalues for the Helmholtz operator in a regular bounded open set Ω in R m relative to some larger set Ω′ if the latter is altered. It is shown that if the frequency is suitably chosen, then the dominant Fredholm eigenvalues decrease when Ω′ is decreased. This property was so far merely established for the Fredholm eigenvalues for the Laplacian (Kress and Roach, J. Math. Anal. Appl. 55 (1976), 102–111). The results obtained will be applied to improve the convergence of a Neumann-Liouville bounded integral operator series, which serves as a tool in determining the solution of the Dirichlet problem.