A set $𝒮=\{x_1,...,x_n\}$ of $n$ distinct positive integers is said to be gcd-closed if $(x_i,x_j)\in 𝒮$ for all $1\le i,j\le n$. Shaofang Hong conjectured in 2002 that for a given positive integer $t$ there is a positive integer $k\left(t\right)$ depending only on $t$, such that if $n\le k\left(t\right)$, then the power LCM matrix $\left({[x_i,x_j]}^t\right)$ defined on any gcd-closed set $𝒮=\{x_1,...,x_n\}$ is nonsingular, but for $n\ge k\left(t\right)+1$, there exists a gcd-closed set $𝒮=\{x_1,...,x_n\}$ such that the power LCM matrix $\left({[x_i,x_j]}^t\right)$ on $𝒮$ is singular. In 1996, Hong proved $k\left(1\right)=7$ and noted $k\left(t\right)\ge 7$ for all $t\ge 2$. This paper develops Hong’s method and provides a new idea...

There are many relations involving the geometric means $G_n\left(x\right)$ and power means ${\left[A_n\left(x^{\gamma }\right)\right]}^{1/\gamma }$ for positive $n$-vectors $x$. Some of them assume the form of inequalities involving parameters. There then is the question of sharpness, which is quite difficult in general. In this paper we are concerned with inequalities of the form $(1-\lambda )G_n^{\gamma }\left(x\right)+\lambda A_n^{\gamma }\left(x\right)\ge A_n\left(x^{\gamma }\right)$ and $(1-\lambda )G_n^{\gamma }\left(x\right)+\lambda A_n^{\gamma }\left(x\right)\le A_n\left(x^{\gamma }\right)$ with parameters $\lambda \in ℝ$ and $\gamma \in (0,1).$ We obtain a necessary and sufficient condition for the former inequality, and a sharp condition for the latter. Several applications of our results are...

Suppose that $A$ is an $n\times n$ nonnegative matrix whose eigenvalues are $\lambda =\rho \left(A\right),{\lambda }_2,...,{\lambda }_n$. Fiedler and others have shown that $det(\lambda I-A)\le {\lambda }^n-{\rho }^n$, for all $\lambda >\rho$, with equality for any such $\lambda$ if and only if $A$ is the simple cycle matrix. Let $a_i$ be the signed sum of the determinants of the principal submatrices of $A$ of order $i\times i$, $i=1,...,n-1$. We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: $det(\lambda I-A)+{\sum }_{i=1}^{n-1}{\rho }^{n-2i}\left|a_i\right|{(\lambda -\rho )}^i\le {\lambda }^n-{\rho }^n$, for all $\lambda \ge \rho$. We use this inequality to derive the inequality that: ${\prod }_2^n(\rho -{\lambda }_i)\le {\rho }^{n-2}{\sum }_{i=2}^n(\rho -{\lambda }_i)$. In the spirit of a celebrated conjecture...