TITLE: An inequality involving $L^1$ and $L^\infty$ norms of a function of a real variable and its derivative
QUESTION [5 upvotes]: I got to the following inequality by a (hopefully correct) tortuous argument: 

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

REPLY [5 votes]: For question 2: assume w.l.o.g. $F$ nondecreasing. If $F(a)\ge 0$
$$\|F'\|_1^2=(F(b)-F(a))^2\le F(b)^2-F(a)^2=\int_a^b 2FF'\le 2\|F\|_1\|F'\|_\infty.$$
Similarly if $F(b)\le 0$. If $F$ changes sign, take $c\in[a,b]$ such that $F(c)=0$ and apply the above to $F$ restricted to $[a,c]$ and $[c,b]$ (using the fact that $(A+B)^2\le 2A^2+2B^2$ and the additivity of the $L^1$-norm).
For equality to occur, $F$ must change sign and we must have $|F'|=\|F'\|_\infty$ a.e. on the two intervals where F is nonzero. Also, the $L^1$-norm of $F$ must be the same on the two intervals, so the equality case is precisely the one mentioned in the question.