Denote $A\circ B$ as the Hadamard product of two matrices, that is, $$A\circ B=(a_{ij}b_{ij}).$$ Let $A$ be a $n\times n$ symmetric positive definite matrix. First, I know that $$tr(A\circ A)\leq tr(A^2)$$ because $$tr(A\circ A)=\sum_{i=1}^na_{ii}^2\leq\sum_{i,j=1}^na_{ij}^2=tr(A^2).$$ In addition, denote $u$ to be the diagonal vector of $A$ and $v$ be its square, that is, $$u=(a_{11},a_{22},\cdots,a_{nn})^T,\quad v=(a_{11}^2,a_{22}^2,\cdots,a_{nn}^2)^T$$
Now, I want to deal with fourth-order items, that is, $tr\left((A\circ A)A^2\right)$, $tr(A^2uu^T)$, $tr(Auv^T)$, $tr\left((A\circ A)^2\right)$, etc. So, can we bound all these fourth-order items by $tr(A^4)$? When generating symmetric positive definite matrix randomly by computer, I always find it true. But I can not prove it!