I have come across an inequality which confuses me: Suppose $X$ has a Normal$(0,\sigma^2)$ distribution. Then
$$ (\mathbb{E}|X|^q)^{1/q} \leq \text{const.} \sqrt{q} \sigma $$
for $q\geq 1$.
I am having trouble figuring out how to show this. I know that the Normal$(0,\sigma^2)$ distribution has moments
$$ \mathbb{E}|X|^q = \sigma^q \frac{2^{\frac{q}{2}}\Gamma(\frac{q+1}{2})}{\sqrt{\pi}}, $$ for all non-negative integers $q$. I think I am missing an upper bound for the Gamma function. Any help?