Some results on matrix monotone functions

AbstractLet A and B be Hermitian matrices. We say that A⩾B if A−B is nonnegative definite. A function ƒ:(0,∞) → R is said to be matrix monotone (m.m.) if A⩾B⩾0 implies that ƒ(A) ⩾ ƒ(B). Matrix convex and concave functions can be similarly defined. In this paper, several known results (some in sharper forms) are given alternative proofs. These include the following: (a) A positive function ƒ(t) is m.m. iff tƒ(t) is m.m. (b) If ƒ(t) is m.m., then both tƒ(t) and ƒ(t)t are matrix convex and are almost always strictly convex; on the other hand, ƒ(t) is always concave and almost always strictly concave. The concavity of m.m. functions and an inequality of Hansen are consequences of a more general inequality on m.m. functions. Some complementary inequalities are also established. We then prove an extension of Lyapunov's theorem. Let A and P be positive definite matrices and either ƒ or 1ƒ be a m.m. function; then the unique matrix solution X of the matrix equation AX + XA = ƒ(A)P + Pƒ(A) is positive definite. A similar result holds for a related equation. Most of the results are obtained by exploiting the integral representation of m.m. functions