  We consider conformally invariant energies $W$ on the group
$\operatorname{GL}^+(2)$ of $2\times2$-matrices with positive determinant, i.e.
$W\colon\operatorname{GL}^+(2)\to\mathbb{R}$ such that \[W(AFB) = W(F)
\qquad\text{for all }\; A,B\in\{aR\in\operatorname{GL}^+(2) \,|\,
a\in(0,\infty)\,,\; R\in\operatorname{SO}(2)\}\,,\] where
$\operatorname{SO}(2)$ denotes the special orthogonal group, and provide an
explicit formula for the (notoriously difficult to compute) quasiconvex
envelope of these functions. Our results, which are based on the representation
$W(F)=h(\frac{\lambda_1}{\lambda_2})$ of $W$ in terms of the singular values
$\lambda_1,\lambda_2$ of $F$, are applied to a number of example energies in
order to demonstrate the convenience of the eigenvalue-based expression
compared to the more common representation in terms of the distortion
$\mathbb{K}:=\frac12\frac{\lVert F\rVert^2}{\det F}$. Special cases of our
results can be obtained from earlier works by Astala et al. and Yan.
