In my work, I keep coming across the term

$$ f(x) = \frac{1}{\sqrt{2\pi}} 2^{\frac{x-1}{2}} \Gamma(\frac{x+1}{2}), $$

in particular for $x \in [0,1]$. I have been working on bounding the density function of a product of powers of standard Gaussians, i.e.

$$ Y = X_1^{e_1}\cdots X_k^{e_k} $$

for $X_i \sim \mathcal{N}(0,1)$ i.i.d., and have been applying those bounds to bound e.g.

$$ E[\lvert Y \rvert^{\delta}], $$

for $\delta \in [0,1]$.

Has anyone else come across this kind of term before? And could you help me understand/prove why, in the range of $ x \in [0,1]$, it is bounded by $\frac{1}{2}$?

Thank you!

Edit: Added more detail, clarified the question, added the missing constant. Thank you for the feedback.

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