Sharp harmonic-mean inequalities for Neumann and Aharonov--Bohm spectra

We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov--Bohm Neumann spectra on surfaces. If $Ω\subset\mathbb S^2$ is smooth, simply connected and proper, then the harmonic mean of $μ_2(Ω)$ and $μ_3(Ω)$ is bounded above by the first positive Neumann eigenvalue of the equal