A Payne-Weinberger inequality for quantum dot Dirac operators

In this work we prove a Payne-Weinberger type inequality for quantum dot Dirac operators defined on bounded and simply connected planar domains. This is a sharp upper bound for their first positive eigenvalue depending only on the isoperimetric deficit of the domain. To this end, we use a recently studied connection with the so-called $\bar{\mathrm{\partial }}$-Robin Laplacian and we establish an analogous inequality for its first eigenvalue, relying on the corresponding inequality for the Robin Laplacian.