New upper and lower bounds for the eigenvalues of the Sturm-Liouville problem

We derive new inequalities for the eigenvaluesλ _k of the Sturm-Liouville problem?u″+qu=λu, u(0)=u(π)=0 under the usual hypothesis thatq has mean value zero. The estimates give upper and lower bounds of the form |λ _k ?k ²|≤p _1,m k ^?m +P _2,m k ^2m ,k ²≥3‖q‖ _m ,m=1, 2 where ‖q‖ _m is the norm ofq in a Sobolev spaceH ^m (0, π) and theP's are homogeneous polynomials of degree at most 3 in ‖q‖ _m . Such bounds are used in the analysis of the inverse Sturm-Liouville problem.