The Hilbert-Schmidt norm, $||A||_F= (\sum_{j=1}^n a_{i,j}^2)^{1/2}$ is clearly always larger than $||A||_{max}$ and is also submultiplicative.

Hence, $||AB||_{max} \leq ||AB||_F \leq ||A||_F ||B||_F$

The Hilbert-Schmidt norm, $||A||_F= (\sum_{i,j=1}^n a_{i,j}^2)^{1/2}$ is clearly always larger than $||A||_{max}$ and is also submultiplicative.

Hence, $||AB||_{max} \leq ||AB||_F \leq ||A||_F ||B||_F$