Given $h:\mathbb{R}\to \mathbb{R}^+$, I want to know whether $f(t)\to 0$ as $t\to \infty$ where $f(t)$ is defined in terms of its Laplace transform $F(s)$: $$F(s)=\frac{\langle h, z_s\rangle}{1-\langle h^2, z_s\rangle}$$
with
$$z_s(i)=\frac{1}{s-2h(i)}$$
Empirically it appears that $\int \mathrm{d}i\ h(i)\le 1$ implies $\lim_{t\to\infty}f(t)=0$, can this be shown more rigorously?
Motivation: $F(s)$ describes evolution of continuous mean-field model solved here