It's a problem from my own :
Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$\Big(\frac{1}{3}\Big)^{\frac{1}{27}}\leq a\Big(b^{abc}\Big)+b\Big(c^{abc}\Big)+c\Big(a^{abc}\Big)=f(a,b,c)\leq 1$$
The idea was to create an inequality where $f(a,b,c)$ takes his values in a very little interval because we have $\Big(\frac{1}{3}\Big)^{\frac{1}{27}}-1<-0.0398$
Furthermore I create this where all the classical method fails .I think to Am-Gm , Jensen , Tchebytchev and so on ...
My first idea was to use rearrangement inequality but it doesn't works too .
Thirdly my idea was to use the Lagrange multiplier but I'm skeptical about this method .
An other point of view is to put $abc=\alpha=constant$ and study the behavior of $f(a,b,c)$.
That's all for me if you have a hint it would be great .
Thanks a lot for sharing your time and knowledge .
Edit
As pointed out by WE tutorial school The RHS is obvious but what happend for the LHS ?