A definitive majorization result for nonlinear operators

Let $\mathfrak{g}$ be a Garding-Dirichlet operator on the set S(n) of symmetric $n\times n$ matrices. We assume that $\mathfrak{g}$ is $I$-central, that is, $D_I\mathfrak{g}=kI$ for some $k>0$. Then $$\mathfrak{g}(A{)}^{\frac{1}{N}}\ge \mathfrak{g}(I{)}^{\frac{1}{N}}(\det A{)}^{\frac{1}{n}}\mathrm{\forall }A>0.$$ From work of Guo, Phong, Tong, Abja, Dinew, Olive and many others, this inequality has important applications.