Range of the first three eigenvalues of the planar Dirichlet Laplacian
  We conduct extensive numerical experiments aimed at finding the admissible range of the ratios of the first three eigenvalues of a planar Dirichlet Laplacian. The results improve the previously known theoretical estimates of M Ashbaugh and R Benguria. We also prove some properties of a maximizer of the ratio $\lambda_3/\lambda_1$. 
Michael Levitin and Rustem Yagudin