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Improvements of Classical Eigenvalue Inequalities I: Quantitative Hersch-deficit and the Joint Spectral Range on the Two-Sphere
For area-$4π$ metrics on the two-sphere, we obtain quantitative refinements of classical eigenvalue inequalities and study the joint behavior of the first two Laplace eigenvalues. We prove a sharp quadratic improvement of Hersch's reciprocal-sum inequality in terms of the first eigenvalue, together
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