Let f:{-1,1}^n -> {-1,1}$$f:\{-1,1\}^n \to \{-1,1\}$$ be a monotone , odd ( f(-x)=-f(x) $f(-x)=-f(x)$) Boolean function.
Let F:[0,1]->[0,1]$$F:[0,1]\to[0,1]$$ denote the probability that f(x1,...,xn)$f(x_1,...,x_n)$ where x1,...xn$x_1,...,x_n$ are i.i.d. +-1$\pm1$ R.V. with probability p$p$ to be 1.
Our question is the following:
Is it true that F$F$ is convex in [0,1/2]$[0,1/2]$ for any such monotone odd f$f$?
(It is easy to see that F$F$ is symmetric around 1/2$1/2$.)
Another way to view this is that dF/dp$dF/dp$ is increasing in [0,1/2]$[0,1/2]$, meaning the bits become more influential as p$p$ is closer to 1/2$1/2$.
So far, in the example we worked out (such as majority) this turned out to be the case, and we could not find any references, nor managed to prove it or find a counterexample. A related fact - when I encountered "graphs" of the probability that certain monotone graph properties are satisfied in G(n,p)$G(n,p)$ as a function of p$p$, they were always "drawn" as convex till the threshold and concave afterwards, but I could not find any explicit references to this phenomena.
