How can I show that $$\left|\sin \frac{s}{2}\right| \geq \frac{|s|}{\pi}$$ $s \in [- \pi , \pi]$, using that $\psi : x \mapsto \sin x$ is a concave function on $[0 , \pi]$?
By definition of concave function, $$ \psi(t \, x + (1 - t) \, y) \geq t \, \psi(x) + (1 - t) \, \psi(y) $$ for each $x , y \in [0 , \pi]$ and for all $t \in [0 , 1]$. My thought was obtain that using just the definition but I think it is not possible but I am not sure. I have drown that and it's obvious but I want to prove that analytically. Can you help me, please? Thank you very much.
Let $f(x)=\sin\frac{x}{2}$.
Thus, $f$ is a concave function on $[0,\pi]$ and we are done
because graph of $y=\frac{x}{\pi}$ is under graph of $f$