Some results on matrix monotone functions
A realvalued function \(f\) defined on the nonnegative reals is called matrix monotone if for all \(n\) and all Hermitian \(n\times n\) matrices with \(0\leq B\leq A\) then \(f(B)\leq f(A)\). The matrix monotone functions were characterized by \textit{C. Löwner} [Math. Z. 38, 177-216 (1934; Zbl 0008.11301)] via an integral representation. Using this the author gives alternative proofs of some of the known results in this area, including the arithmetic mean/harmonic mean inequality, \textit{F. Hansen}'s inequality [Math. Ann. 246, 249-250 (1984; Zbl 0407.47012)] and some of the authors own results, and he extends some of these. The author extends Lyapunov's classical theorem and shows: Let \(A\) and \(P\) be positive definite. If \(f\) or \(1/f\) is matrix monotone then the solution \(X\) of \(AX+XA=f(A)P+Pf(A)\) is positive definite.
arithmetic mean/harmonic mean inequality
scientific article; zbMATH DE number 4112793
scientific article