  The objective of this note is to establish the Determinant Majorization
Formula $F(A)^{1\over N} \geq \det(A)^{1\over n}$ for all operators $F$
determined by an invariant Garding-Dirichlet polynomial of degree $N$ on
symmetric $n \times n$ matrices. Here "invariant" means under the group O$(n)$,
U$(n)$ or Sp$(n)$ when the matrices are real symmetric, Hermitian symmetric, or
quaternionic symmetric respectively. This greatly expands the applicability of
the recent work of Guo-Phong-Tong and Guo-Phong for differential equations on
complex manifolds. It also relates to the work of Abja-Olive on interior
regularity.
  Further applications to diagonal operators and to operators depending on the
ordered eigenvalues are given. Examples showing the preciseness of the results
are presented. For the application to Abja-Olive's work, and other comments in
the paper, we establish some results for Garding-Dirichlet operators in
appendices. One is an exhaustion lemma for the Garding cone. Another gives
bounds for higher order derivatives, which result from their elegant
expressions as functions of the Garding eigenvalues. There is also a discussion
of the crucial assumption of the Central Ray Hypothesis.
