4 problems

Let $A,B\geq 0$, and let $p,q,r$ be real parameters satisfying $0<p\leq q$ and $r\geq 0$. Here $\succ_{\log}$ denotes log-majorization. Reverse log-majorization conjecture. One has…

Generalized geometric-mean conjecture. If $0\leq t\leq1$, $p\geq1$, and either $r,s\geq1$ or $r,s\leq0$, then

Lemos–Soares conjecture. If $A,B\geq0$, $r,s\in\mathbb{R}$, and $0\leq t\leq1$, then

Let $A$ and $B$ be the positive definite matrices appearing in the two inequalities of; namely, the inequalities compare the corresponding trace expressions involving $A$, $B$, and…