  The main result of this paper is a sharp upper bound on the first positive
eigenvalue of Dirac operators in two dimensional simply connected $C^3$-domains
with infinite mass boundary conditions. This bound is given in terms of a
conformal variation, explicit geometric quantities and of the first eigenvalue
for the disk. Its proof relies on the min-max principle applied to the squares
of these Dirac operators. A suitable test function is constructed by means of a
conformal map. This general upper bound involves the norm of the derivative of
the underlying conformal map in the Hardy space $\mathcal{H}^2(\mathbb{D})$.
Then, we apply known estimates of this norm for convex and for nearly circular,
star-shaped domains in order to get explicit geometric upper bounds on the
eigenvalue. These bounds can be re-interpreted as reverse Faber-Krahn-type
inequalities under adequate geometric constraints.
