Prove that $(-1)^n \text{Laguerre}_n(2) \leq 1$. I would like to prove the following inequalities on Laguerre polynomials evaluated at point 2:
$$
(-1)^n \text{Laguerre}_n(2) \leq 1
$$
This seems to hold numerically.
I tried to use the recurrence relation between Laguerre polynomials but I was not successful. Any ideas? I also tried expression via Bessel function $J_0$ and then contour integral but it was not successful.
 A: Theorem 8.22. on Szego's Orthogonal polynomials the asymptotics of the Laguerre polynomials (called there Fejer's formula) looks like
$$L_n(x)=\frac{e^{x/2}}{2^{1/4}\sqrt{\pi}\cdot n^{1/4}}cos \left ( 2\sqrt{nx}-\pi /4\right )+O(n^{-3/4}),\, \text{for }x>0.$$
Take $x=2,$ we have then that
$$|L_n(2)|\leq \frac{e}{2^{1/4}\sqrt{\pi}}\left |\frac{cos(2^{3/2}\sqrt n-\pi /4)}{n^{1/4}} \right |+\left |O(n^{-3/4})\right |\leq 1.2896\cdot \frac{1}{n^{1/4}}+\frac{C}{n^{3/4}},$$
for some constant $C.$
Even if the constant is something big, you can check the result for small values of $n.$
Here is a graph using $C=2$ which I would say is big according to simulation:

Also, this is a very good survey about the whole problem
A: 
This is a graph of $(-1)^n L_{n}^{(0)}(2)$ where $0 \leq n \leq 2500$. In this range $|(-1)^n L_{n}^{(0)}(2)| \leq 1$. This may be of interest. The Digital Library of Mathematical Functions, NIST gives $| L_{n}^{(0)}(2)| \leq e$ in equation 18.14.8 as above. Proving a better bound is probably difficult.
