Monotonicity of rotationally invariant convex and rank 1 convex functions

Let f : M-nxn --> R boolean OR {infinity} be a function on the space M-nxn of a by n matrices that is invariant with respect to the left and right multiplication by proper orthogonal tensors. It is shown that f(A) less than or equal to f((A) over bar) if f is convex and the partial sums of the singular values of A, (A) over bar is an element of M-nxn satisfy certain ordering inequalities. The same holds if f is rank 1 convex and the partial products of the singular values satisfy analogous inequalities. The proofs emphasize the roles of the ordered-forces inequalities and the Baker-Ericksen inequalities for invariant convex and rank 1 convex functions. As an application, the evaluation of the convex and lamination convex hulls of fully rotationally invariant sets by Dacorogna and Tanteri is simplified and similar results are given for sets invariant only with respect to the proper orthogonal group.