About a Proposition on Escobar's paper “The Geometry of the First Non-zero Stekloff Eigenvalue”

We expect to prove that the scope of inequality $\nu_{1}(g)\geq\Big(\max_{x\in\partial M}e^{-f(x)}\Big)\nu_{1}(g_{0})$ is not as broad as the result of Proposition $2$ suggests in . We aim to show that this inequality cannot be strict by proving that the inequality $Q_{g}(\phi) \geq\Big(\displaystyle\max_{\partial M}e^{-f}\Big)Q_{g_{0}}(\phi)$ is in fact an equality, and that this equality only occurs if the function $f$ is constant on $\partial M$; this would suggest that the lower bound in Example $7$ in may need to be more carefully stated or perhaps revised and that the conclusion of Proposition $4$ in may not be correct. Similarly, more general applications of this inequality may need to be handle with care since, as it is showed, the result is not as general as it seems to be proposed.