it's an inequality by me :
Let $a,b,c>0$ such that $a+b+c=1$ and $a\geq b \geq c $ then we have : $$\sum_{cyc}\cos(a\cos(b))\geq \sum_{cyc}\cos(a\cos(c))$$
This inequality is very precise . I try Karamata's inequality because the function $f(x)=\cos(x)$ is concave on $[0,1]$ but we have with $a\geq b \geq c$ : $$a\cos(b)\leq a\cos(c)$$
So we are in the wrong way .
I try also approximation with (for $x\in(0,1)$ ):
$$1-\frac{x^2}{2}+\frac{x^4}{24}\geq\cos(x)\geq 1-\frac{x^2}{2}$$
Remains to show :
$$\sum_{cyc}\Big(1-\frac{(a\cos(c))^2}{2}+\frac{(a\cos(c))^4}{24}\Big)\leq \sum_{cyc}\Big(1-\frac{(a\cos(b))^2}{2}\Big)$$
I think this last inequality is true but now I'm stuck...
I would like a hint if you have.
Thanks a lot for sharing your time and knowledge .