Bounding the modified Bessel function of the first kind

Question

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.

Answer 1

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 $x > 0$.

Yudell L. Luke, Inequalities for generalized hypergeometric functions, Journal of Approximation Theory, Volume 5, Issue 1, January 1972, pp. 41–65.

(Link to the article on ScienceDirect)