I want to prove that the following function is concave (as a part of another proof).
$$f(p) = \max_{\begin{matrix}x,y\\0\le x \le 1\\0\le y \le 1 \\ x * y = p\end{matrix}} \lambda h(x) + \bar{\lambda}h(y)$$
In which $\bar{a}=(1-a)$ and $a*b=a\bar{b}+\bar{a}b$
and $0\le p \le 1$
and $\lambda$ is a constant $0 \le \lambda \le 1$.
and $h(x)=-x\log_2(x)-\bar{x}\log_2(\bar{x})$ is the binary entropy function (which is concave).
I have numerically calculated $f(p)$ for different values of $\lambda$ and it seems to be concave. I have tried directly applying the concave function definition, use of convexity preserving operations, and calculating hessian to prove concavity, but I couldn't prove it.