lower bound of modified Besselfunction

Question

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument.

There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.

Answer 1

This cannot be valid for $c,\alpha>0$ and all positive $x$, because for $x\rightarrow 0$ the LHS approaches $c$ while the RHS approaches $0$ for $\nu > 0!$ You have $$\frac{(\tfrac{1}{2}x)^{\nu}}{\Gamma(\nu+1)} \le I_{\nu}(x)$$ for all $x,\nu > 0$ and asymptotic expansions for large $x:$ $$I_{\nu}(x) \sim \frac{e^x}{\sqrt{2\pi x}}\left(1 + O(x^{-1})\right) $$