  In this note we prove two isoperimetric inequalities for the sharp constant
in the Sobolev embedding and its associated extremal function. The first such
inequality is a variation on the classical Schwarz Lemma from complex analysis,
similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini,
and Ransford, while the second generalises an isoperimetric inequality for the
first eigenfunction of the Laplacian due to Payne and Rayner.
