This question is a small part of a bigger problem I am working on. Let $h(p)$ be the binary entropy function. That is, for $p \in (0,1)$ $$h(p) = -p\log_2(p) - (1-p)\log_2(1-p)$$
Define the convolution of $p$ and $q$ $\quad 0\leq p,q\leq 1$ as $$p*q = p(1-q) + q(1-p)$$
Now consider the function $f:[0,1/2)\to\mathbb{R}$ $$f(x) = \frac{h(p*x) - h(x)}{1-h(x)}$$
I need to show its minimum value is $4p(1-p)$. When I tried the usual differentiation method, I was unable to isolate the variables. Hence I am looking for an alternate method. Here are some observations that may prove useful:
- $h(p*x)$ is a monotone increasing function for $x \in [0,1/2)$. It varies from $h(p)$ to $1$.
- $h(x)$ is symmetric about $x=1/2$, is a concave function and achieves its maxima at $x=1/2$ where it equals $1$.
- $*$ is commutative and associative. Also, $1-p*x = (1-p)*x$
- $4p(1-p) = 2(p*p)$
Let me know your thoughts on approaching this. If you need more information, let me know in comments. Thanks.
Edit: Bonus: If we/I are able to resolve this, I will include the main problem as part of my answer.