  In this paper, we prove an isoperimetric inequality for lower order
eigenvalues of the Dirichlet Laplacian on bounded domains of a Euclidean space
which strengthens the well-known Ashbaugh-Beguria inequality conjectured by
Payne-P\'olya-Weinberger on the ratio of the first two Dirichlet eigenvalues
and makes an important step toward the proof of a conjecture by
Ashbaugh-Benguria.
