I'm interested in proving the uniqueness of the intersection of the following two functions for all $L<0$:
$$f_{L}(K) = \sqrt{1 -\frac{2K}{K^{2} - L^{2}}}$$
and
$$g_{L}(K) = V\Big((K+L) \sqrt{1 -\frac{2K}{K^{2} - L^{2}}}\Big)$$
where
$$V(x) = \frac{\int_{0}^{2\pi} \cos\theta \,\mathrm{e}^{x \cos\theta}\mathrm{d}\theta}{\int_{0}^{2\pi} \,\mathrm{e}^{x \cos\theta}\mathrm{d}\theta}$$
on the interval $K\in (L + \sqrt{1-L^{2}}, \infty)$. Now these functions have some nice properties: They both go to zero on the left of the interval, they both approach one on the right and they are non decreasing and concave on the interval with continuous derivatives. Furthermore, their first derivatives diverge on the left of the interval in such a way that I know $$f_{L}(L + \sqrt{1-L^{2}} + \epsilon)>g_{L}(L + \sqrt{1-L^{2}} + \epsilon)$$ for some small $\epsilon>0$ (i.e. they are ordered close to the left). On the other hand for large $K$
$$f_{L}(K)\sim 1 - 1/K$$
while
$$g_{L}(K)\sim 1-1/2K$$
from which I am tempted to conclude that they intersect at least once. Since these are very specific functions, asking for a complete proof might be a bit much but any hints for a proof would be appreciated. I have also taken a look at various posts on this site of questions I thought might be similar but none have been of help.