In studying the solution to a coupled second order ODE, I noticed, based on numerics, that the equation
$$K_0(x) - bK_0(ax) = 0$$
where $a, b > 1$ appears to only have one solution for $x > 0$. Since the ratio
$$\frac{K_0(x)}{K_0(ax)}$$
seems to be strictly monotonically increasing (which would prove that there is only one solution), I wanted to show that the derivative of this ratio is positive. This leads to showing that
$$a\frac{K_0(x)}{K_1(x)} - \frac{K_0(ax)}{K_1(ax)} > 0$$
which would be true if the function
$$\frac{K_0(x)}{K_1(x)}$$
converges to $0$ as $x \to 0$ and is concave. Both conditions appear to be true numerically. I can probably prove the first condition using asymptotics, but I'm stumped on the second condition. It appears that the ratio of Bessel functions is a well-studied subject, but so far I haven't found anything regarding the concavity of the ratio of these two Bessel functions. Alternatively one can just show directly that the function is subadditive (I think subadditivity is a weaker condition), but I also didn't find anything about that. The closest I can find is conjecture 2 of this paper, but that conjecture is stronger than what I need, so I'm hoping there is some other weaker results out there that shows it.
Does anyone know of some good resources/has an idea how to prove it?
