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.
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