  In this paper, we prove two results for the Robin eigenvalue problem. One is
an upper bound for the ratio of the first two eigenvalues which can be used to
recover the PPW conjecture proved by M.S.Ashbaugh and R.D.Benguria, the other
is a reverse Holder inequality for the first eigenfunction which is a natural
generalization of Chiti's reverse Holder inequality for the first eigenfunction
of Dirichlet Laplacian.
