Upper bound on the modified Bessel function of the first kind

Question

I am looking for an upper bound tighter than $e^x$ for the modified Bessel function of the first kind. Does there exist $\epsilon\in (0,1)$ such that $I_0(x)\le e^{x^{1-\epsilon}}$ for all $x\ge 0$ ?

Answer 1

No, it does not exist. In fact, we have the asymptotic, $$I_0(x) \sim \frac{e^x}{\sqrt{2\pi x}}$$ as $x \rightarrow +\infty$, so we can't get any improvement on the exponent.