# About the exponential bounds for modified Bessel function

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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The Abramowitz-Stegun Handbook has some asymptotics that may help you. – Zsbán Ambrus Dec 31 '10 at 17:31
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 $M$. – Suvrit Dec 31 '10 at 18:57

## 1 Answer

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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Thank you very much, I think your comments will help me a lot! – user11936 Dec 31 '10 at 18:43
A user named "lonewolf" has pointed out that the link is now dead. – S. Carnahan Jun 23 '13 at 23:00