i'm looking for an upper bound for the modified Bessel function of the first kind of a +ive real argument. It seems that it satisfies the inequality :
$$I_{n}(x)\leqslant \frac{x^{n}}{2^{n}n!}e^{x}$$
But i'm not able to prove this.
[Math] Bounding the modified Bessel function of the first kind
approximationasymptoticsinequalityspecial functions
Best Answer
It was proved by Yudell L. Luke in 1972 that
$$ 1 < \Gamma(\nu+1)\left(\frac{2}{x}\right)^\nu I_\nu(x) < \cosh x $$
for $x > 0$ and $\nu > -1/2$. This implies your inequality since
$$ \cosh x - e^x = -\sinh x < 0 $$
for $x > 0$ and hence
$$ \cosh x < e^x $$
for $x > 0$.