Let $p,q$ be positive probability densities on a measure space $(E,\mathcal E,\mu)$, $\lambda$ denote the Lebesgue measure on $\mathcal B(\mathbb R)$, $d\in\mathbb N$, $\varphi:[0,1)^d\to E$ be bijective and $\left(\mathcal B([0,1))^{\otimes d},\mathcal E\right)$-measurable with $\left.\lambda\right|_{[0,\:1)}^{\otimes d}\circ\varphi^{-1}=p\mu$ and $f:E\to[0,\infty)^3$ be $\mathcal E$-measurable.
I want to find the minimum value (not necessarily the minimizer itself) of $$F_i(h):=\int\frac{|hf_i|^2}{p\left(q\circ\varphi^{-1}\right)}\:{\rm d}\mu$$ over all $i\in\{1,2,3\}$ and a finite collection $\mathcal H$ of bounded $\mathcal E$-measurable $h:E\to[0,\infty)$. How can we obtain this value (at least numerically)?