It is given that: $\left |\det(A) \right |\leq n^{\frac{-n}{2}}\left \| A \right \|^{n}$ where $A$ is an $n$ by $n$ matrix, and $\left \| A \right \|$ is the Hilbert Schmidt norm (i.e: $\left \| A \right \|=\left ( \sum_{i,j=1}^{n}a_{ij}^{2} \right )^{\frac{1}{2}}$).
Now, I want to prove the following inequality based on the above one:
$\left |\det(A) \right |\leq \prod_{j=1}^{n}\left ( \sum_{i=1}^n a_{ij}^2\right )^{\frac 12}.$
Any help?