inequality based on Hermite polynomial

Question

How to prove that $$|H_n(x)| \leq |H_n(ix)|?$$

I have tried with the explicit representation of the Hermite polynomial, but can't reach the target. Any clue please.

Answer 1

Hint : it just has to do with the fact that $H_n(x) = \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^kc_{n, k} x^{n-2k}$ for some $c_{n, k}\geq 0$.