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.
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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 $c_{n, k}\geq 0$.
Sary
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$H_n(ix)=\sum_{k=0}^{[n/2]} (-1)^k C_{n,k}(ix)^{n-2k}=i^{n}\sum_{k=0}^{[n/2]} C_{n,k}x^{n-2k}$ – user3575652 Mar 03 '15 at 04:07
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Yes ? What happens if x is positive ? Negative ? – Sary Mar 03 '15 at 12:32
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So, is it $i^n H_n(x)=H_n(ix)$ or not? – Boby Mar 01 '16 at 17:38
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I know this is a very old post, but has anyone a proof or a reference for the fact illustrated by Sary? – Davide F. Nov 27 '16 at 21:36