A B if A − B is positive semidefinite.

Given a strictly increasing function f: (a, b) → R, we define the partial order _f on the set of Hermitian matrices with spectrum contained in (a, b) by

A _fB if f(A) f(B).

We say that the partial orders and _f are equivalent ona set of Hermitian matrices if

A B if and only if A _fB for all A,B ε .

It is clear that if the cone is commutative, i.e., AB = BA for all A, B ε , then the two partial orders are equivalent. Stepniak conjectured the converse for the function f(t) = t², and proved it for n 3. We provide a counterexample to Stepniak's conjecture for n 4, and we characterize the convex cones of positive semidefinite matrices on which and _f are equivalent for a class of functions that includes f(t) = t^p, p> 1, and f(t) = e^t. We introduce the class of strongly monotone matrix functions and prove the following result of independent interest: Let ƒ be a strongly monotone matrix function of order n, and suppose that A and B are n-by-n Hermitian matrices such that A − B is positive semidefinite and that A and B have no common eigenvectors. Then f(A) − f(B) is positive definite. We also show that the functions f(t) = t^p, 0 < p < 1, and f(t) = log t are strongly monotone of all orders. We also consider the partial order _t². In this special case it is possible to obtain the results using more elementary techniques. We prove some results about the partial orders _t^p and _exp. We conclude with two open questions.