Thanks Bruno, Sorry to bother you with the title, I do not have a lot of experience in posting problems. In fact this is a research problem and I am not aware of any place that a solution to it exists. I am sure that the claim is true, since I have simulated the recursion for large values of $n$ and it is true. Many thanks again.

Hi, Many thanks for your comment. The problem is that the range of $2x-x^2$ over $[0, \frac 12]$ is $[0, \frac 34]$. As a result what you mentioned does not work here.