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 $\mathrm{\Omega }\subset {\mathbb{S}}^2$ is smooth, simply connected and proper, then the harmonic mean of ${\mu }_2(\mathrm{\Omega })$ and ${\mu }_3(\mathrm{\Omega })$ is bounded above by the first positive Neumann eigenvalue of the equal-area geodesic disk, with equality only for disks. For simply connected surfaces with Gaussian curvature bounded above, we obtain the magnetic analogue for the first two Aharonov–Bohm eigenvalues. The proof combines a two-dimensional reciprocal Rayleigh–Ritz principle with Green-level comparison. A key additional ingredient is a spectral ordering theorem for magnetic spherical caps: for $0<\nu <1\mathrm{/}2$, the first two eigenvalues lie in the angular sectors of effective orders $\nu$ and $1-\nu$. We prove this by the factorization $L_0=T^{\ast }T$, $L_1=T{T}^{\ast }$ and an exact Neumann–Dirichlet spectral shift. We also obtain sharp full-sphere and closed-surface bounds, and an annular inequality in terms of conformal modulus and flux.