  In an Euclidean Jordan algebra V of rank n, an element x is said to be
majorized by an element y, if the corresponding eigenvalue vector of x is
majorized by the eigenvalue vector of y in R^n. In this article, we describe
pointwise majorization inequalities of the form `T(x) majorized by S(x)', where
T and S are linear transformations induced by Schur products. Specializing, we
recover analogs of majorization inequalities of Schur, Hadamard, and
Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as
majorization inequalities connecting quadratic and Lyapunov transformations on
V. We also show how Schur products induced by certain scalar means (such as
arithmetic, geometric, harmonic, and logarithmic means) naturally lead to
majorization inequalities.
