We disprove Laugesen's conjecture that the ratio of the first two Robin eigenvalues decreases with the positive boundary parameter. On any prescribed compact interval of positive parameters, the ratio is strictly increasing on a suitable connected polyhedral domain in every dimension at least two; in the plane, the dom...
We prove P\'olya's inequality for every Neumann eigenvalue of every Euclidean circular annulus. The proof preserves the exact Weyl area term through a radial action identity and combines variational bounds with uniform Bessel-phase and angular-sum estimates. It is entirely analytic when the width is at most one quarter...
We prove both inequalities in Laugesen and Siudeja's Conjecture~6.34 for the first two positive Neumann eigenvalues of triangles. The equilateral triangle uniquely maximizes their harmonic mean at fixed perimeter and their geometric mean at fixed area. Two-dimensional trial spaces reduce the proof to polynomial positiv...
We prove Laugesen and Siudeja's conjecture on Neumann eigenvalues of isosceles triangles. After multiplication by the squared diameter, the first positive symmetric eigenvalue increases strictly with the apex angle $\theta$, while the first antisymmetric eigenvalue decreases strictly up to the equilateral triangle and...
