As well-known the modified Bessel functions of orders zero and one of the first kind, $I_0(x)$ and $I_1(x)$ are positive and increasing functions over $x\in(0, \infty)$, while the modified Bessel functions of orders zero and one of the second kind, $K_0(x)$ and $K_1(x)$ are positive and decreasing functions over $x\in(0, \infty)$.
How can I prove the following inequality with $\alpha>1$ and $k>0$?
$$I_0(k\alpha)K_1[k(\alpha-1)]-K_0(k\alpha)I_1[k(\alpha-1)]>0.$$
In other words, how to prove $I_0(x)K_1(y)>K_0(x)I_1(y)$ with $x>y>0$?
I believe the inequality is true as can be observed by plotting. Thank you very much!
Best Answer
Introduce the function $$ f(x,t)=I_0(x+t)K_1(x)-K_0(x+t)I_1(x). $$ We want to show that $f(x,t)>0$ for all $x>0,t\geq 0$.
Consider first the case $f(x,0)$. We can prove that $f(x,0)>0$ by writing $$ f(x,0)=(I_0(x)-I_1(x))K_1(x)+I_1(x)(K_1(x)-K_0(x)) $$ and using the monotonicity properties (see here: https://dlmf.nist.gov/10.37) $$ I_0(x)>I_1(x)\mbox{ and }K_1(x)>K_0(x)\mbox{ for all }x>0. $$
Next, by a direct computation, we have $$ \frac{\partial}{\partial t}f(x,t) =I_1( t + x) K_1(x) + I_1(x)K_1(t + x) $$ which is manifestly positive for $x>0,t\geq 0$. Hence $$ f(x,t)=f(x,0)+\int_0^t (I_1( \tau+ x) K_1(x) + I_1(x)K_1(\tau + x))\mathrm d \tau>0. $$