Sharp Hyperbolic Cutoffs and Dimension-Sharp Counterexamples for Reverse Araki-Type Inequalities

We study reverse Araki-type trace inequalities and log-majorizations beyond the exponent $2$.

For arbitrary nonnegative nondecreasing weights, we show that $s=2$ is the sharp dimension-free boundary: for every $s>2$, explicit one-parameter $3\times3$ positive definite examples violate the reverse Liu--Cheng trace inequality and the corresponding dual formulation of Shi--Wei--Wang, whereas the reverse inequality remains valid for every $s\geq1$ in dimension $2$.

For power weights, a larger region survives and is bounded by a sharp hyperbola.

In normalized variables, for $s>2$, \[ A^{r+s}B^s \succ_{\log}A^r (A^{1/2} BA^{1/2})^s \] holds for all positive semidefinite matrices in every finite dimension if and only if $0\leq r\leq s/(s-2)$; beyond this range, even the associated trace inequality fails for $3\times3$ positive definite matrices.

Equivalently, for $0<p\leq q$ and $q>2p$, the sharp condition is $0\leq r\leq pq/(q-2p)$.

Combined with the known all-$r$ regime $p\leq q\leq2p$, this completes the reverse log-majorization phase diagram.