A negative exponent range for Audenaert's complementary McCarthy trace inequality

Abstract

Audenaert introduced a class of complementary McCarthy type trace inequalities in his work on completely monotone functions and Bernstein functions; the same problem was later included in the problem list of Audenaert and Kittaneh. The known results cover the corresponding directions for $q\le -2$, $0<q\le 1$, $1\le q\le 2$, and $2\le q\le 3$, while the negative range $-2<q<0$ was left as a conjectural case. We prove the negative exponent inequality, in fact for every $q<0$. After the inversion $X=A^{-1}$, $Y=B^{-1}$, the problem reduces to a trace inequality for the parallel sum $X:Y$. The proof uses the Kubo--Ando mean chain, an Ando--Hiai type log-majorization for matrix geometric means, and a finite-dimensional Schatten H\"older inequality, including the quasi-norm range. We also record that the positive exponent side $q\ge 3$ follows directly from Audenaert's norm-compression inequality for positive semidefinite $2\times2$ block matrices, and in fact holds for all $q\ge 2$. Finally, we determine the equality case on the negative side: equality holds if and only if $A=B$.