The value $l>0$ is bounded above by either of the following inequalities with $a,c>0$,
$$l \leq \max \{a, b\}$$ or $$l \leq \max\{c, d\}.$$
My though was $$l \leq \max \{\sqrt{ac},\sqrt{ad},\sqrt{bc},\sqrt{bd} \}, \quad l \geq0.$$ Is it correct? Can we found a tight upper bound of $l$?
Note: I am interested mostly for the case where $b=d=1$. However, a more general intuition on how we handle such problem is highly appreciated.