Let $0< s_1 < s_2$ and $0<Z_1<Z_2$, consider functions $F_j:[0, \infty)\to [0, \infty)$ defined by $$F_j(t) = \frac{1}{Z_j} I_0(2\sqrt{ts_j}),$$ where $j = 1,2$ and $I_0$ is the modified Bessel function of the first kind. I want to show $F_1$ and $F_2$ intersects only once.
Attempt: Using $I_0(0)=1$, we know $F_1(0) > F_2(0)$. Using large argument asymptotic for $I_0$ and $s_1 < s_2$, it is easy to see $F_2$ is much larger than $F_1$ for large $t$, so an intersection must exist.
I tried to calculate the derivative for the difference, but unfortunately the difference is not monotone for all choices of the parameters. Are there other ways to pursue this?
Thank you