3 problems

Let $A$ be an $n\times n$ Hermitian positive semidefinite matrix, let $\Pi_\mu(A)$ be its μ-Schur power matrix, and let $\mu\in[0,1]$. Bapat–Lal's conjecture. The largest eigenvalu…

Let $A$ be a positive definite matrix, and let $P_\mu(A)$ denote its μ-permanent. de Falco's conjecture. There exists $\epsilon<-1$ such that $P_\mu(A)$ is a strictly increasing fu…

Let $A$ be an $n\times n$ Hermitian positive definite nondiagonal matrix, and let $P_\mu(A)$ denote its μ-permanent. Bapat–Lal's conjecture. $P_\mu(A)$ is a strictly increasing fun…