On some isoperimetric inequalities involving eigenvalues of fixed membranes

In 1956, Hersch (1965) derived some isoperimetric inequalities for eigenvalues of a fixed membrane ΩΩ, simply connected, with a center of symmetry O. In this note we are going to derive some sharper versions of Hersch’s results. More precisely, if ΩΩ is symmetric of order 2 we show that we have λ2(Ω)+λ3(Ω)≤2λ2(Dro)λ2(Ω)+λ3(Ω)≤2λ2(Dro), where DroDro is a disc of radius ro(Ω)ro(Ω) (the conformal radius of ΩΩ at O). Also, if ΩΩ is symmetric of order 4, we have λ4(Ω)+λ5(Ω)≤2λ4(Dro)λ4(Ω)+λ5(Ω)≤2λ4(Dro).