On a Question of Bhatia and Jain III

Let \(p_1,p_2,\ldots ,p_n,\;(n\ge 2),\) $p_1,p_2,\dots ,p_n,(n\ge 2),$ be distinct positive numbers and \(r>0\) $r>0$ . We propose to study a comparison of the positivity properties of two families of matrices, \(K_{r+1}=\begin{bmatrix}\frac{p_i^{r+1}+p_j^{r+1}}{p_i+p_j}\end{bmatrix}\) $K_{r+1}=\left[\begin{array}{c}\frac{p_i^{r+1}+p_j^{r+1}}{p_i+p_j}\end{array}\right]$ and \(B_r=\begin{bmatrix}{|p_i-p_j|^r}\end{bmatrix}\) $B_r=\left[\begin{array}{c}|p_i-p_j{|}^r\end{array}\right]$ in full. Indeed, Bhatia and Jain (Spectr. Theory 5(1):71–87, 2015) studied about \(B_r\) $B_r$ and carried out rigorous analysis on the study of \(K_{r}\) $K_r$ . They conjectured therein that inertia of \(K_{r+1}\) $K_{r+1}$ and \(B_{r}\) $B_r$ are same for all \(r>0\) $r>0$ . We settle a congruence relation between these two families in this paper.