Given $A_1, ..., A_n$ ($n\geq 3$), where each $A_i$ is a $d$-by-$d$ symmetric, positive definite matrix, define $S = A_1\cdot A_2\cdot...\cdot A_n$ (product of all the $A_i$'s). Let $\lambda_1(A)$ and $\lambda_d(A)$ be the top and bottom eigenvalues of $A$, respectively.

Can we say anything about $\lambda_{d}( S + S^T )$? I know we can lower bound $\lambda_{d}(S+S^T) \geq -2 \lambda_{1}(A_1)\cdot ... \cdot \lambda_{1}(A_n)$ but can we say anything less conservative involving the $\lambda_{d}(A_i)$'s? Ideally, I'd like to be able to say something like $\lambda_{d}(S+S^T) \geq 2\lambda_{d}(A_1)\cdot ... \cdot \lambda_{d}(A_n)$ but I have a feeling this is not true.