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Let Ω be an open connected subset of R n for which the Poincaré inequality holds. We consider the Dirichlet eigenvalue problem for the Laplace operator in the open subset φ(Ω) of R n , where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then we show Lipschitz type inequalities for the reciprocals of the eigenvalues of the Rayleigh quotient φ(Ω) |Dv| 2 dy φ(Ω) |v| 2 dy upon variation of φ, which in particular yield inequalities for the proper eigenvalues of the Dirichletdoi:10.4171/zaa/1240 fatcat:kqysk6gsljbohku5e6zx24zi6m