Some moths ago I stumbled upon the following upper and lower bound on the modified Bessel function of the first kind and zeroth order:
$\frac{e^{x}}{1 + 2x} < I_0(x) < \frac{e^{x}}{\sqrt{1 + 2x}}, x > 0$.
Now I have trouble finding the reference, is there any way of finding this or proving these bounds? Are they even true?
I found paper I was looking for!
Yang, Zhen-Hang; Chu, Yu-Ming, On approximating the modified Bessel function of the first kind and Toader-Qi mean, J. Inequal. Appl. 2016, Paper No. 40, 21 p. (2016). ZBL1332.33009.https://link.springer.com/article/10.1186/s13660-016-0988-1
The double inequality I was looking for (and proof) can be found in Theorem 3.1.
FYI this lower bound can be easily improved to gain a factor of $\sqrt{x}$ to become only a constant factor from the correct $x\to\infty$ asymptotics. See Lower bounds on the modified Bessel function of the first kind
