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Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that $$\sum_{i,j}(G^5)_{ij}\le\sum_{i,j}(G^3)_{ij}\,?$$

This is true if it is additionally assumed that all off-diagonal entries of $G$ are the same.

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If I have not committed any mistake, please, find below a counter-example.

Counter-example. Let $G\in \mathbb{S}^3_{+}$ be defined by $$ G = \begin{pmatrix} 1 & -\frac{2}{5} & 0 \\ -\frac{2}{5} & 1 & -\frac{2}{5}\\ 0 & -\frac{2}{5} & 1 \end{pmatrix}. $$

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    $\begingroup$ $\frac{2}{5}$ replaced with $\frac{1}{2}$ works too. $\endgroup$
    – Peter Mueller
    Commented Jun 3, 2023 at 11:27
  • $\begingroup$ Mark, thank you for the edit. @PeterMueller, indeed. And it works for 1/3 as well, but it fails (i.e., the inequality holds true), e.g., for $1/5$. Not clear at this point for me what would be a 'good' sufficient condition for this inequality to hold. $\endgroup$
    – Augusto Santos
    Commented Jun 3, 2023 at 12:50
  • $\begingroup$ Thank you for your answer. $\endgroup$
    – Iosif Pinelis
    Commented Jun 4, 2023 at 3:09
  • $\begingroup$ @IosifPinelis, thank you. $\endgroup$
    – Augusto Santos
    Commented Jun 4, 2023 at 12:42

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