Inequalities for Eigenvalues of a General Clamped Plate Problem

Let D be a connected bounded domain in R-n. Let 0 < mu(1) <= mu(2) <= ... <= mu(k) <= ... be the eigenvalues of the following Dirichlet problem: {Delta(2)u(x) + V(x)u(x) = mu rho(x)u(x), x is an element of D u vertical bar(partial derivative D) = partial derivative u/partial derivative n vertical bar(partial derivative D) = 0, where V(x) is a nonnegative potential, and rho(x) is an element of C(<(D)over bar>) is positive. We prove the following inequalities: mu(k+1) <= 1/k Sigma(k)(i=1) mu(i) + [8(n + 2)/n(2) (rho(max)/rho(min))(2)](1/2) x 1/k Sigma(k)(i=1)[mu(i)(mu(k+1) - mu(i))](1/2), n(2)k(2)/8(n + 2) <= (rho(max)/rho(min))(2) [Sigma(k)(i=1) mu(1/2)(i)/mu(k+1) - mu(i)] x Sigma(k)(i=1) mu(1/2)(i).