I have on domain $[0,\infty)$ a known and positive function $f(x)$ and two unknown functions $g(x), h(x)$ that start positive when $x=0$.

I also know that if $h(x)$ is positive, then $g(x)$ is also positive.

I define the induction

$$h(x)=\int_0^x f(t)g(t)h(t) dt$$

I imagine that is possible to say that $h(x)$ (and so $g(x)$) remains positive, but I could not find a formal argument.

Thank you in advance.