A small Lemma for the Poincare inequality

Question

Let $[a,b]$ be a compact intervall $f(x):=\sin(\frac{x-a}{b-a}\pi)$ and $z\in [a,b]$ a zero of the function $f$, $a\leq x \leq b$ and $U=$conv{$x,z$} with "conv" meaning the convex hull. Why does it hold that $$ f(x)=\int_z^xf'(y)dy \Rightarrow |f(x)|\leq \int_U|f'(x)| $$ I don't understand a) Why the convex hull "U" is used. b) Why "$\Rightarrow$" holds.

Answer 1

$U$ just means the interval with endpoints $x$ and $z$. Since we don’t know whether $x<z$ or not, we can’t write it as an interval $[x,z]$ or $[z,x]$.

The second part is due to triangle inequality for integrals. From the assumption, we know that $$|f(x)|=\left|\int_z^xf’(y)\,dy\right|=\left|\int_Uf’(y)\,dy\right|\leq \int_U |f’(y)|\,dy.$$

Thanks to @amsmath for guiding!