Concave functions and positive block matrices

Abstract

Let $f\left(t\right)$ be a nonnegative concave function on $[0,\infty )$. For a positive (semidefinite) matrix partitioned into four blocks such that $AX=XA$ or $BX=XB$, we prove that$\Vert f\left(\left[\begin{array}{cc}A& X\\ X^{⁎}& B\end{array}\right]\right)\Vert \le 2\Vert f\left(\frac{A+B}{2}\right)\Vert$ for all unitarily invariant norms. The case $X=0$ is already new, contains two classical trace inequalities due to Rotfel'd and von Neumann, and generalizes an important basic majorization. Our proof is based, and also extends, a theorem of Bourin and Mhanna involving the width of the numerical range of X. For Schatten q-quasinorms, $0<q<1$, and nonnegative convex functions vanishing at 0, we obtain the reverse inequality.