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Dear colleagues, I want to know if there are some results on the bounds of modified Bessel functions $I_\alpha(x)$ and $K_\alpha(x)$? Especially, I need the exponential bounds for them, that is to say if it is possible to get a result like

$|I_\alpha(x)|\leq C exp(b x), x\geq 0$

where $C$ is a constant or a polynomial of $x$, b is a constant, $\alpha$ is a real number. Also the inequality of the same form about $K_\alpha(x)$.

Thank you in advance!

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  • $\begingroup$ The Abramowitz-Stegun Handbook has some asymptotics that may help you. $\endgroup$
    – Zsbán Ambrus
    Commented Dec 31, 2010 at 17:31
  • $\begingroup$ Maybe this is going the wrong way round, but another possible bound can be obtained by using $I_\nu(z) = (z/2)e^z M(0.5+\nu; 1+2\nu; 2z) / \Gamma(\nu+1)$, where $M$ is Kummer's function. Now, you can invoke Stirling's approximation for $\Gamma$, and one of the many approximations for $M$. $\endgroup$
    – Suvrit
    Commented Dec 31, 2010 at 18:57

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There are some bounds of that form in this paper. See also the first reference at the end of the paper.

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  • $\begingroup$ Thank you very much, I think your comments will help me a lot! $\endgroup$
    – user11936
    Commented Dec 31, 2010 at 18:43
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    $\begingroup$ A user named "lonewolf" has pointed out that the link is now dead. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 23, 2013 at 23:00

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