Duality for Garding operators and a new inequality of GM/AM type

This article concerns Garding Dirichlet operators, and also the theory of Garding polynomials. Garding Dirichlet operators are those which arise from homogeneous polynomials on the space S(n) of symmetric $n\times n$ polynomials, which are hyperbolic with respect to any positive definite matrix. (In particular, any polynomial on ${\mathbb{R}}^n$, which is hyperbolic with respect to elements in $({\mathbb{R}}_{+}{)}^n$, gives rise to a Garding operator on S(n), in many ways.) Here a new duality theorem is developed with many special cases and examples. The dual $g^{\ast }$ of a Garding operator $g$ is used to prove a new basic inequality for $g$. An important Lemma of Guo-Phong-Tong is improved upon as a final application.