The upper bound and lower bound of $\left\| A \right\|_{F}^{2} + \left\| B \right\|_{F}^{2} $

Question

If $a$, $b$ are both real numbers, then the inequality $2ab \leq a^2 + b^2$ holds true; If $\mathbf{a}$, $\mathbf{b}$ are both column vectors, then the inequality $2\mathbf{a}^{T}\mathbf{b} \leq ||\mathbf{a}||^2_2 + ||\mathbf{b}||^2_2$ is also true, where $T$ means transpose operation and $||\cdot||_{2}$ is the l2-modulus of the vector. What I want to know is that if $A$ is a $m\times r$ matrix, and $B$ is a $n\times r$ matrix, does there exist a constant $c$ that makes the following form of inequality true? $$ c\times ||AB^T||_{F}^2 \leq ||A||_{F}^2 + ||B||_{F}^2, $$ or in the opposite form, is there a constant $d$, such that the following form of inequality holds true? $$ ||A||_{F}^2 + ||B||_{F}^2 \leq d\times ||AB^T||_{F}^2. $$ Here $||\cdot||_{F}$ means the frobenius norm.