Let $x, \nu \geq 0$.
By WolframAlpha I found that $$K_\nu(x) \leq \frac{1}{x^{\nu}}$$ is an upper bound. I want to know, can this upper bound be improved? Where can I find such properties? Thanks
2026-04-01 10:03:23.1775037803
Bessel K function upper bound $K_\nu(x) \leq \frac{1}{x^{\nu}}$
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Your upper bound does not hold. If $x,\nu\geq 0$, it is true that:
$$ x^\nu K_\nu(x) \leq (2\nu-2)!! = 2^{\nu-1}\Gamma(\nu),$$
(equality is attained when $x\to 0^+$) and that bound can be probably improved by probabilistic techniques, by recalling that the Bessel K function arises as the PDF of a product distribution.