Prove that the following function is decreasing (as a function of a) for a > 0 when 0 < r < 1:
$${K\_2(ar)I\_2(a)-I\_2(ar)K\_2(a)\over I\_2(a)}I\_2(ar).$$


The problem arose in the analysis of a model for yield stress fluids. We have numerical evidence, but I would be interested in an analytical proof.



Monotonicity of a combination of Bessel functions
This discussion makes me even firmer in my opinion that introducing fancy notation for special functions and making long lists of related formulae in reference books makes more harm than good and that we would know much more about and be at more ease with them if everybody had to start from the basics and deal with bare definitions every time he needed to establish some of their properties.


Just write your expression as
$$
I(ra)^2\int\_r^1\left[-\frac{d}{ds}\frac{K(sa)}{I(sa)}\right]\,ds=\int\_r^1\frac{I(ra)^2}{I(sa)^2} a(I'(sa)K(sa)-I(sa)K'(sa))\,ds
$$
Now notice that while finding the solutions of a second order ODE is in general next to impossible, finding their Wronskian is trivial. In particular, for this case we have $ a(I'(sa)K(sa)-I(sa)K'(sa))=\frac{c}s$ where $c$ is some positive number. Once we understand that, all that remains is to establish that $\frac{I(ra)}{I(sa)}$ is decreasing in $a>0$ for $r<s$, which immediately follows from the fact that $I$ is the sum of a Taylor series with non-negative coefficients.


