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Let $\tau_k(\Omega)$ be the $k$-th eigenvalue of the Laplace operator in abounded domain $\Omega$ of the form $\Omega_{\text{out}} \setminus\overline{B_{\alpha}}$ under the Neumann boundary condition on $\partial\Omega_{\text{out}}$ and the Robin boundary condition with parameter $h \in(-\infty,+\infty]$ on the sphere $\partial B_\alpha$ of radius $\alpha>0$centered at the origin, the limiting case $h=+\infty$ being understood as theDirichlet boundary condition on $\partial B_\alpha$. In the case $h>0$, it isknown that the first eigenvalue $\tau_1(\Omega)$ does not exceed$\tau_1(B_\beta \setminus \overline{B_\alpha})$, where $\beta>0$ is chosen suchthat $|\Omega| = |B_\beta \setminus \overline{B_\alpha}|$, which can beregarded as a reverse Faber-Krahn type inequality. We establish this result forany $h \in (-\infty,+\infty]$. Moreover, we provide related estimates forhigher eigenvalues under additional geometric assumptions on $\Omega$, whichcan be seen as Szeg\H{o}-Weinberger type inequalities. A few counterexamples tothe obtained inequalities for domains violating imposed geometric assumptionsare given. As auxiliary information, we investigate shapes of eigenfunctionsassociated with several eigenvalues $\tau_{i}(B_\beta \setminus\overline{B_\alpha})$ and show that they are nonradial at least for allpositive and all sufficiently negative $h$ when $i \in \{2,\ldots,N+2\}$. Atthe same time, we give numerical evidence that, in the planar case $N=2$,already second eigenfunctions can be radial for some $h<0$. The latter factprovides a simple counterexample to the Payne nodal line conjecture in the caseof the mixed boundary conditions.

Reverse Faber-Krahn and Szego-Weinberger type inequalities for annular domains under Robin-Neumann boundary conditions