This function is quite interesting as it reaches a maximum in 0. This can be seen without difficulty computing the first and second derivatives. It is

$$B(a,r)=\frac{K_2(ar)I_2(a)-I_2(ar)K_2(a)}{I_2(a)}I_2(ar)$$

that gives

$$B(0,r)=\frac{1}{4}(1-r^4)>0$$

$$B'(0,r)=0$$

$$B''(0,r)=-\frac{1}{12}r^2(1-r^2)^2<0$$

for the given interval. This means that this function is decreasing for $a>0$. The problem is if there are some other points where the first derivative can become zero changing the concavity of the curve. So, the first derivative has the following involved expression

$$B'(a,r)=\frac{1}{2 I^2_2(a)}\left[I_2(ar)^2 (I_1(a)+I_3(a)) K_2(a)+\right.$$ $$I_2(a) I_2(ar) (-2 r (I_1(ar)+I_3(ar)) K_2(a)+$$ $$I_2(ar) (K_1(a)+K_3(a))+r I_2^2(a) ((I_1(ar)+I_3(ar)) K_2(ar)$$ $$\left.-I_2(ar) (K_1(ar)+K_3(ar)))\right]$$

Now we notice that $I_2$ is a monotonic increasing function that never becomes zero and $K_2$ is a monotonic decreasing function that never becomes zero for the given intervals and similarly is true for $I_1,\ I_3$ and $K_1,\ K_3$. All these functions are positive. So, it is not difficult to realize that the first derivative is a monotonic function and never hits zero again being just a balance of functions never reaching zero unless $a=0$ and having monotonic behavior, $K_i$ are decreasing functions and $I_i$ increasing functions. We also note that

$$\lim_{a\rightarrow\infty}B'(a,r)=0$$

that can be proved using the asymptotic formula for these Bessel functions. Now, combining monotonicity and positivity of these Bessel functions, starting from 0 and reaching asymptotically 0 at increasing values of the argument, they can just reach an extremum and never cross zero again. This can also be seen with a simple plot

evaluated at $r=0.1,0.3,0.5,0.7,0.9$.

This function is quite interesting as it reaches a maximum in 0. This can be seen without difficulty computing the first and second derivatives. It is

$$B(a,r)=\frac{K_2(ar)I_2(a)-I_2(ar)K_2(a)}{I_2(a)}I_2(ar)$$

that gives

$$B(0,r)=\frac{1}{4}(1-r^4)>0$$

$$B'(0,r)=0$$

$$B''(0,r)=-\frac{1}{12}r^2(1-r^2)^2<0$$

for the given interval. This means that this function is decreasing for $a>0$. The problem is if there are some other points where the first derivative can become zero changing the concavity of the curve. So, the first derivative has the following involved expression

$$B'(a,r)=\frac{1}{2 I^2_2(a)}\left[I_2(ar)^2 (I_1(a)+I_3(a)) K_2(a)+\right.$$ $$I_2(a) I_2(ar) (-2 r (I_1(ar)+I_3(ar)) K_2(a)+$$ $$I_2(ar) (K_1(a)+K_3(a))+r I_2^2(a) ((I_1(ar)+I_3(ar)) K_2(ar)$$ $$\left.-I_2(ar) (K_1(ar)+K_3(ar)))\right]$$

Now we notice that $I_2$ is a monotonic increasing function that never becomes zero and $K_2$ is a monotonic decreasing function that never becomes zero for the given intervals and similarly is true for $I_1,\ I_3$ and $K_1,\ K_3$. All these functions are positive. So, it is not difficult to realize that the first derivative is a monotonic function and never hits zero again being just a balance of functions never reaching zero unless $a=0$ and having monotonic behavior, $K_i$ are decreasing functions and $I_i$ increasing functions. We also note that

$$\lim_{a\rightarrow\infty}B'(a,r)=0$$

that can be proved using the asymptotic formula for these Bessel functions. SoNow, this function reaches a maximum to combining monotonicity and positivity of these Bessel functions, starting from 0 and then decreases monotonically toward reaching asymptotically 0 at increasing values of the argument, they can just reach an extremum and never cross zero again. This can also be seen with a simple plot

evaluated at $-\infty$.r=0.1,0.3,0.5,0.7,0.9$. 3 Removed wrong formula already substituted in a preceding edit This function is quite interesting as it reaches a maximum in 0. This can be seen without difficulty computing the first and second derivatives. It is $$B(a,r)=\frac{K_2(ar)I_2(a)-I_2(ar)K_2(a)}{I_2(a)}I_2(ar)$$ that gives $$B(0,r)=\frac{1}{4}(1-r^4)>0$$ $$B'(0,r)=0$$ $$B''(0,r)=-\frac{1}{12}r^2(1-r^2)^2<0$$ for the given interval. This means that this function is decreasing for$a>0$. The problem is if there are some other points where the first derivative can become zero changing the concavity of the curve. So, the first derivative has the following involved expression $$B'(a,r)=\frac{1}{2I_2^2(a)}\left[\frac{4}{a}I_2^2(ar)I_2(a)K_2(a)+\right.$$ $$\left.I_2(a)I_2(ar)(-2rI_1(ar)+I_3(ar))K_2(a)+\frac{4}{a}I_2(ar)K_2(a)\right].$$ $$B'(a,r)=\frac{1}{2 I^2_2(a)}\left[I_2(ar)^2 (I_1(a)+I_3(a)) K_2(a)+\right.$$ $$I_2(a) I_2(ar) (-2 r (I_1(ar)+I_3(ar)) K_2(a)+$$ $$I_2(ar) (K_1(a)+K_3(a))+r I_2^2(a) ((I_1(ar)+I_3(ar)) K_2(ar)$$ $$\left.-I_2(ar) (K_1(ar)+K_3(ar)))\right]$$ Now we notice that$I_2$is a monotonic increasing function that never becomes zero and$K_2$is a monotonic decreasing function that never becomes zero for the given intervals and similarly is true for$I_1,\ I_3$and$K_1,\ K_3$. All these functions are positive. So, it is not difficult to realize that the first derivative is a monotonic function and never hits zero again being just a balance of functions never reaching zero unless$a=0$and having monotonic behavior,$K_i$are decreasing functions and$I_i$increasing functions. We also note that $$\lim_{a\rightarrow\infty}B'(a,r)=0$$ that can be proved using the asymptotic formula for these Bessel functions. So, this function reaches a maximum to 0 and then decreases monotonically toward$-\infty\$.

2 Expanded explanation and corrected first derivative
1