Eigenvalues of a Sturm-Liouville problem and inequalities of Lyapunov type

We consider the eigenvalue problem u " + lambda u + p(x)u = 0 in (0, pi), u(0) = u(pi) = 0, where p is an element of L-1(0, pi) keeps a fixed sign and \\p\\(L1) > 0, and we obtain some lower and upper bounds for \\p\\(L1) in terms of its nonnegative eigenvalues lambda. Two typical results are: (1) \\p\\(L1) > root lambda\sin root lambda\ if lambda > 1 and is not the square of a positive integer; (2) \\p\\(L1) less than or equal to 16/pi if lambda = 0 is the smallest eigenvalue.