Consider the following Sturm Liouville problem on an interval $[a,b]$
$$\frac{\mathrm{d}}{\mathrm{d} x}\left[p(x) \frac{\mathrm{d} y}{\mathrm{d} x}\right]+q(x) y=-\lambda w(x) y$$ for given coefficient function $p(x),q(x),w(x)>0$. Here we can assume they are smooth. One can consider two sets eigenvalues $$0<\lambda_{1}^D<\lambda_{2}^D<\lambda_{3}^D<\cdots<\lambda_{n}^D<\cdots \rightarrow \infty$$ $$\lambda_{1}^N<\lambda_{2}^N<\lambda_{3}^N<\cdots<\lambda_{n}^N<\cdots \rightarrow \infty$$ where $\lambda_i^D$ are Dirichlet boundary condition and $\lambda_i^N$ are Neumann boundary condition. It is quite easy to show $\lambda_i^N\leq \lambda_i^D$ by the variational characterizations of the eigenvalues. However, it seems that under some condition of $p,q,w$, we will have $$\lambda_{i+1}^N\leq \lambda_i^D$$ I encounter this when reading some papers, and it says this is well known. Does anyone know a good reference for this type of inequality? Who first found this? Do you have a simple proof?
PS. I googled a little bit and found many works on eigenvalue inequality of this type for Laplacian operator on dimension greater than 1. I am just interested in one dimension and here the operator is more general than Laplacian.
