It's the last post of the day so :
Let $0<x<1$ then we have : $$(1-x^{4(1-x)^2}-(1-x)^{4x^2})^{\frac{6}{5}}\geq \Big(\frac{\cos(\cos(\cos(\cos(2\pi x))))-\cos(\cos(\cos(\cos(2\pi ))))}{2}\Big)^2$$
I try series expansion but it becomes awful .
I know that :
For $0<x<1$ : $$(1-x^{4(1-x)^2}-(1-x)^{4x^2})\geq 0$$ The proof is here
The equality case are $x=0$ and $x=1$ and $x=0.5$
Maybe this could be useful .
Maybe we can generalize this with the Dottie number.
I have not the time to continue so ...
Thanks a lot for sharing your time and knowledge .
