AI 中文总结
本文证明了Audenaert互补McCarthy迹不等式对所有负指数q<0成立，运用Kubo–Ando均值链等工具，确定等号条件，还补充了正指数侧的相关结论。
AI 中文摘要
Audenaert在其关于完全单调函数和Bernstein函数的研究中引入了一类互补McCarthy型迹不等式；同一问题后续被纳入Audenaert与Kittaneh的问题列表。已知结果涵盖了q≤-2、0<q≤1、1≤q≤2及2≤q≤3的对应方向，而负范围-2<q<0被留作猜想情形。我们证明了该负指数不等式，实际上对所有q<0均成立。经逆变换X=A⁻¹、Y=B⁻¹后，问题简化为平行和X:Y的迹不等式。证明运用了Kubo–Ando均值链、矩阵几何均值的Ando–Hiai型对数优化，以及有限维Schatten Hölder不等式（含拟范数范围）。我们还指出，正指数侧q≥3可直接由Audenaert关于半正定2×2分块矩阵的范数压缩不等式导出，且实际上对所有q≥2均成立。最后，我们确定了负侧的等号情形：当且仅当A=B时等号成立。
英文摘要
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 Hölder 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 $A=B$.
