Magnetic spectral bounds on starlike plane domains

We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that $\sum_{j=1}^n \Phi \big( \lambda_j A/G \big)$ is maximal for a disk whenever $\Phi$ is concave increasing, $n \geq 1$, the domain has area $A$, and $\lambda_j$ is the $j$-th Dirichlet eigenvalue of the magnetic Laplacian $\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2$. Here the flux $\beta$ is constant, and the scale invariant factor $G$ penalizes deviations from roundness, meaning $G \geq 1$ for all domains and $G=1$ for disks.