Polyconvexity implies Hill's inequality in {rm SL}(2)

For compressible nonlinear isotropic elasticity it is well known that rank-one convexity, polyconvexity and the monotonicity of the Cauchy stress tensor with respect to the logarithmic stretch tensor (the true stress-true strain monotonicity, TSTS-M$^+$) are independent constitutive conditions which should, however, all together be satisfied for a physically meaningful description of idealized elastic materials. In the incompressible case, TSTS-M$^+$ turns into Hill's inequality since the Cauchy stress $\sigma$ reduces to the Kirchhoff stress $\tau$. Hill's inequality requires then monotonicity of the Kirchhoff stress in terms of the logarithmic stretch tensor evaluated for incompressible response. In this paper we clarify how the a priori independent notions of Legendre-Hadamard ellipticity (LH), polyconvexity and Hill's inequality are nevertheless intimately connected. More precisely, by providing several alternative proofs, we show that both LH-ellipticity (rank-one convexity) and polyconvexity imply the weak Hill inequality in the incompressible two-dimensional case.