Let \(D\) be the unit disc. In this remarkably elegant paper the authors prove that \(\| Sf\|\leq 2\) for any convex domain \(G= f(D)\) (\(\| Sf\|\) is the Schwarzian norm of \(f\)). Moreover, if \(\| Sf\|= 2\), no convex function \(f\) maps \(D\) onto a quasidisk. The authors use the fact that \(Sf= {2\varphi'\over (1-\varphi)^2}\) for \(|\varphi|\leq 1\), for convex \(f\). Earlier, Nehari considered approximations of convex domains using the Schwarz-Christoffel formula applied to a convex polygon. The proof of Nehari is complicated and his analysis is not precise. In addition, the authors prove that every bounded convex map has the Schwarzian norm \(\| Sf\|< 2\) which is a surprising result.