Let $A$ be a $m \times r$ matrix and $B$ be a $r \times n$ matrix, I wonder if there exists an inequality like the following: $$ \left \| AB \right \|_F \leq f(m,r,n)g(A,B) , $$ or $$ \left \| AB \right \|_F \ge f(m,r,n)g(A,B) , $$ where $f(m,r,n)$ is a function of the dimension of matrix $m,r,n$, and $g(A,B)$ is a function of $A$ or $B$. For example, does $\left \| AB \right \|_F \leq mr^2n \left \| A \right \|_F$ (here $f(m,r,m) = mr^2n$ is a function of $m$, $r$ and $n$, and $g(A,B) = \left \| A \right \|_F$ is a function of $A$) ?
I looked through the matrix reference books, but could not find any satisfactory answer, only some less relevant ones like this link. Does anyone know the answer?
The Frobenius norm is sub-multiplicative since it is an operator norm. Meaning: $$\|AB\|_F \leq \|A\|_F\|B\|_F.$$ Is this what you are looking for? I think this wikipedia article contains good explanations for this.