An Araki-Lieb-Thirring inequality for geometrically concave and geometrically convex functions (Q1947106)

Denoting by \({\mathcal P}_n\) the space of Hermitian positive definite \(n\times n\) matrices, let \(A,B\in{\mathcal P}_n\). Define the log-submajorization relation \(\lambda(A)\prec_{w(\log)}\lambda(B)\) by weak majorization of logarithms of the spectra \(\lambda(A)\) and \(\lambda(B)\). The Araki-Lieb-Thirring inequality (\textit{H.~Araki} [Lett. Math. Phys.~19, No. 2, 167--170 (1990; Zbl 0705.47020)], \textit{E.~H.~Lieb} and \textit{W.~E.~Thirring} [Stud. math. Phys., Essays Honor Valentine Bargmann, 269--303 (1976; Zbl 0342.35044)]) states that if \(0<t\leq 1\), then \(\lambda(A^tB^t)\prec_{w(\log)}\lambda^t(AB)\), while the reverse holds if \(t\geq 1\). More generally, consider a continuous nonnegative function~\(f\) defined on an interval~\(I=[0,x_0)\) with \(x_0>0\) (possibly infinite). The author proves that \(\lambda(f(A)f(B))\prec_{w(\log)}f^2(\lambda^\frac{1}{2}(AB))\) for all \(A,B\in{\mathcal P}_n\) with \(\lambda(A),\lambda(B)\subset I\) if and only if \(f\) is geometrically concave (i.e. \(\sqrt{f(x)f(y)}\leq f(\sqrt{xy})\) for all \(x,y\in I\)) and \(0\leq xf'(x)\leq f(x)\) for all \(x\in I\) where \(f'(x)\) exists. He also proves that the reverse holds if and only if \(f\) is geometrically convex (i.e. \(\sqrt{f(x)f(y)}\geq f(\sqrt{xy})\) for all \(x,y\in I\)) and \(xf'(x)\geq f(x)\) for all \(x\in I\) where \(f'(x)\) exists. As an application, the author provides a complementary inequality to the Golden-Thompson inequality.