On inequalities between Hermitian and symmetric forms

AbstractIf A and C are n x n Hermitian matrices and if B is an n x n symmetric matrix, we consider inequalities of the following type: x∗Ax⩾∣xTBx∣ or x∗Ax⩾∣x∗Cx∣ for all complex n-vectors x. Necessary and sufficient conditions for these inequalities to hold are derived, and it is shown that the set of pairs of such matrices is a convex cone which is closed under composition by the Schur product. Infinite divisibility, which is a continuous analog of Schur composition, is studied. Related results involving the domination of a bilinear form by pairs of quadratic forms are given. The origin of these ideas in complex function theory is discussed, as are applications to probability theory and harmonic analysis