For a positive semi-definite $d\times d$ matrix $A$, $ (\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $ for every $S\in\text{SPD}_{d}$.

Question

For a positive semi-definite $d\times d$ matrix $A$, $$ (\text{det}(AS))^{\frac{1}{d}}\leq\frac{1}{d}\text{Tr}(AS) $$ for every $S\in\text{SPD}_{d}$.

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Answer 1

$$\left(\Pi_{i}\lambda_{i}^{AS}\right)^{\frac{1}{d}}\leq\frac{1}{d}\left(\Sigma_{i}\lambda_{i}^{AS}\right)$$

The above inequality is done with the assumptions.