Answer by Giulio R for How to prove the inequality including Bessel functions?
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...
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In other words, you want to prove that, for the worst case,$$f(x)=I_0(x)\, K_1(x)-I_1(x)\, K_0(x)~>~0$$It is simple if you consider small and large values of $x$.Using their asymptotic...
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How to prove the inequality including Bessel functions?
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...
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