Abstract
Let \(p_1,p_2,\ldots ,p_n,\;(n\ge 2),\) be distinct positive numbers and \(r>0\). We propose to study a comparison of the positivity properties of two families of matrices, \(K_{r+1}=\begin{bmatrix}\frac{p_i^{r+1}+p_j^{r+1}}{p_i+p_j}\end{bmatrix}\) and \(B_r=\begin{bmatrix}{|p_i-p_j|^r}\end{bmatrix}\) in full. Indeed, Bhatia and Jain (Spectr. Theory 5(1):71–87, 2015) studied about \(B_r\) and carried out rigorous analysis on the study of \(K_{r}\). They conjectured therein that inertia of \(K_{r+1}\) and \(B_{r}\) are same for all \(r>0\). We settle a congruence relation between these two families in this paper.
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Kapil, Y., Mandeep & Singh, M. On a Question of Bhatia and Jain III. Results Math 79, 51 (2024). https://doi.org/10.1007/s00025-023-02064-5
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DOI: https://doi.org/10.1007/s00025-023-02064-5