Inequalities between Dirichlet and Neumann eigenvalues of vibrating strings

The Dirichlet eigenvalues ({{lambda_{n}}_{n=1}^{infty}}) and Neumann eigenvalues ({{mu_{n}}_{n=1}^{infty}}) of the string equation ({varphi'' (x) +lambda rho (x) varphi(x) =0}) are considered. It is known that ({ mu_{n} < lambda_{n} < mu_{n+2}}) for all n. The purpose of this paper is to provide conditions on the mass density ({rho(x)}) under which ({lambda_{n} < mu_{n+1}}) or ({mu_{n+1} < lambda_{n}.})