I got to the following inequality by a (hopefully correct) tortuous argument:

If $F:[a,b] \to \mathbb{R}$ is a absolutely continuous

monotonefunction then: $$ \|F'\|_1^2 \leq 4 \|F\|_1 \, \|F'\|_\infty $$

**Question 1:** Does this inequality have a name?

**Question 2:** Is there a short proof?

**Remark**: Cases of equality should be with $\pm F$ like that:

(The plateau in the middle may vanish; in this case $F$ is affine with zero mean.)