Extremals for eigenvalues of Laplacians under conformal mapping (Q1266252)

\textit{G. Pólya} and \textit{G. Szegő} [``Isoperimetric inequalities in mathematical physics'', Princeton, NJ (1951; Zbl 0044.38301)] have proved that under a conformal mapping normalization for simply connected plane domain \(\Omega\) the first eigenvalue of Laplace equation with Dirichlet boundary conditions is maximal for a disc \(D,\) i.e. \(\lambda_{1}(\Omega)\leq \lambda_{1}(D).\) Moreover \(\sum_{j=1}^{n}\frac{1} {\lambda_{j}(\Omega)}\geq \sum_{j=1}^{n} \frac{1}{\lambda_{j}(D)}\) for each \(n=1,2,3,\dots\) [see \textit{G. Pólya} and \textit{M. Schiffer}, J. Anal. Math. 3, 245-345 (1954; Zbl 0056.32701)]. The authors prove that for every convex increasing function \(\Phi\) \[ \sum_{j=1}^{n}\Phi \biggl(\frac{1} {\lambda_{j}(\Omega)} \biggr)\geq\sum_{j=1}^{n} \Phi\biggl(\frac{1}{\lambda_{j}(D)}\biggr),\quad n=1,2,3,\dots \] and obtain results of this type for the simply and doubly connected domains on cones, cylinders and surfaces of variable curvature. They prove also that for \(N\)-dimensional Riemannian manifold \(M_{s}\) with boundary and each smooth function \(w\) on the closure of \(M_{g}\) the ``zeta-type'' functional \(\sum_{j=1}^{n}\Phi(\lambda_{j}(w)^{-1})\) is convex with respect to the density \(w\).