  In this note we prove a version of the classical Schwarz lemma for the first
eigenvalues of the Laplacian with Dirichlet boundary data. A key ingredient in
our proof is an isoperimetric inequality for the first eigenfunction, due to
Payne and Rayner, which we reinterpret as an isoperimetric inequality for a
(singular) conformal metric on a bounded domain in the plane.
