I am trying to maximise the function $F(x) = -\int_0^{\infty} p(x)\log(p(x))dx$ across all functions $p: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ such that $p(x) = p(1/x)$ for all $x \in (0, 1]$, and $\int_{\mathbb{R}_{\geq 0}} p(x) dx = 1$. I have simplified the problem via the following:
$\int_0^{\infty} p(x)\log(p(x))\, dx$
$=\int_0^1 p(x)\log(p(x))\, dx + \int_1^{\infty} p(x)\log(p(x))\, dx$
$=\int_0^1 p(x)\log(p(x)) \,dx + \int_0^1 (\frac{1}{y^2})p(y)\,log(p(y))\, dy$
$= \int_0^1 (1+\frac{1}{x^2})p(x)\log(p(x))\,dx$
So now I am trying to minimise $ F(x) = \int_0^1 (1+\frac{1}{x^2})p(x)\log(p(x))\,dx$ across all functions $p:[0,1] \rightarrow \mathbb{R}_{\geq 0}$, $\int_{\mathbb{R}_{\geq 0}} p(x) dx = 1$.
Is this possible? If so, what techniques should I use to do so? I am inexperienced in this area of maths and do not know where to start so any pointers are appreciated, especially if anyone has an idea of how difficult this will be to do. Thanks!