  In this note, we provide an explicit formula for computing the quasiconvex
envelope of any real-valued function $W\colon\operatorname{SL}(2)\to\mathbb{R}$
with $W(RF)=W(FR)=W(F)$ for all $F\in\operatorname{SL}(2)$ and all
$R\in\operatorname{SO}(2)$, where $\operatorname{SL}(2)$ and
$\operatorname{SO}(2)$ denote the special linear group and the special
orthogonal group, respectively. In order to obtain our result, we combine
earlier work by Dacorogna and Koshigoe on the relaxation of certain conformal
planar energy functions with a recent result on the equivalence between
polyconvexity and rank-one convexity for objective and isotropic energies in
planar incompressible nonlinear elasticity.
