Faber–Krahn conjecture for the auxiliary operator H_E

Let $\Omega \subset \mathbb{R}^2$ be a $C^\infty$ simply connected domain, and let $\mu^\Omega(E)$ be the first eigenvalue of the auxiliary operator $H_E^\Omega$. Let $\mathbb{D}$ be the unit disk. Faber–Krahn conjecture for $\mu^\Omega(E)$. For all $E>0$, there holds

$$\mu^\Omega(E) \geq \frac{\pi}{|\Omega|}\mu^\mathbb{D}\Big(\sqrt{\frac{|\Omega|}{\pi}}E\Big).$$

Moreover, there is equality in the above inequality if and only if $\Omega$ is a disk. This is proposed as a new Faber–Krahn-type inequality for the first eigenvalue of $H_E^\Omega$ and is discussed as a conjectural tool for studying the preceding Dirac eigenvalue conjecture. Its resolution status is not specified in the supplied text.