The second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.

The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

Avian malaria, caused by Plasmodium spp., is an emerging disease in New Zealand (NZ). To detect Plasmodium spp. infection and quantify parasite load in NZ birds, a real-time polymerase chain reaction (PCR) (qPCR) protocol was used and compared with a nested PCR (nPCR) assay. A total of 202 blood samples from 14 bird species with known nPCR results were tested. The qPCR prevalences for introduced, native and endemic species groups were 70, 11 and 21%, respectively, with a sensitivity and specificity of 96·7 and 98%, respectively, for the qPCR, while a sensitivity and specificity of 80·9 and 85·4% were determined for the nPCR. The qPCR appeared to be more sensitive in detecting lower levels of parasitaemia. The mean parasite load was significantly higher in introduced bird species (2245 parasites per 10 000 erythrocytes) compared with endemic species (31·5 parasites per 10 000 erythrocytes). In NZ robins (Petroica longipes), a significantly lower packed cell volume was found in birds that were positive for Plasmodium spp. compared with birds that were negative. Our data suggest that introduced bird species, such as blackbirds (Turdus merula), have a higher tolerance for circulating parasite stages of Plasmodium spp., indicating that introduced species are an important reservoir of avian malaria due to a high infection prevalence and parasite load.