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.



An inequality for certain positive-semidefinite matrices
If I have not committed any mistake, please, find below a counter-example.



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


