Let $f(p)$ be a Lebesgue measurable function on $p\in (0,1)$ satisfying $$ f^2(p(1-p))= f(p^2)\times f((1-p)^2). $$ $f(p)=\lambda p^{\alpha}$, for some $\lambda \in \mathbb{R}$ and $ \alpha \in \mathbb{R}$, is a solution.
Question: Is this the only solution of the equation?