We focus on the minimization problem of the eigenvalue ratio for the $p$-Laplacian operator with Robin boundary conditions on an interval $[0,\hat{\pi}]$, where $\hat{\pi}= \frac{2\pi}{p\sin(\pi/p)}$. Using variational techniques and Pr\"ufer-type transformations, we show that the constant weight is not minimizing for the class of concave weights.
