Reverse Faber-Krahn inequalities for Zaremba problems

Let $\Omega$ be a multiply-connected domain in $\mathbb{R}^n$ ($n\geq 2$) of the form $\Omega=\Omega_{\text{out}}\setminus \bar{\Omega_{\text{in}}}.$ Set $\Omega_D$ to be either $\Omega_{\text{out}}$ or $\Omega_{\text{in}}$. For $p\in (1,\infty),$ and $q\in [1,p],$ let $\tau_{1,q}(\Omega)$ be the first eigenvalue of \begin{equation*} -\Delta_p u =\tau \left(\int_{\Omega}|u|^q \text{d}x \right)^{\frac{p-q}{q}} |u|^{q-2}u\;\text{in} \;\Omega,\; u =0\;\text{on}\;\partial\Omega_D, \frac{\partial u}{\partial \eta}=0\;\text{on}\; \partial \Omega\setminus \partial \Omega_D. \end{equation*} Under the assumption that $\Omega_D$ is convex, we establish the following reverse Faber-Krahn inequality $$\tau_{1,q}(\Omega)\leq \tau_{1,q}({\Omega}^\bigstar),$$ where ${\Omega}^\bigstar=B_R\setminus \bar{B_r}$ is a concentric annular region in $\mathbb{R}^n$ having the same Lebesgue measure as $\Omega$ and such that (i) (when $\Omega_D=\Omega_{\text{out}}$) $W_1(\Omega_D)= \omega_n R^{n-1}$, and $(\Omega^\bigstar)_D=B_R$, (ii) (when $\Omega_D=\Omega_{\text{in}}$) $W_{n-1}(\Omega_D)=\omega_nr$, and $(\Omega^\bigstar)_D=B_r$. Here $W_{i}(\Omega_D)$ is the $i^{\text{th}}$ $quermassintegral$ of $\Omega_D.$ We also establish Sz. Nagy's type inequalities for parallel sets of a convex domain in $\mathbb{R}^n$ ($n\geq 3$) for our proof.