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 generalizes an
isoperimetric inequality for the first eigenfunction of the Laplacian
due to Payne and Rayner.