Can we prove, without using Laguerre polynomials, that 
$f_n(x)=O(\frac{n!}{\sqrt{n}})$ i.e. that

$$
\exists C>0, \exists N\in\mathbb N, \forall x\geq0, \forall n\geq N :\ \big| f_n(x) \big|\leq C \frac{n!}{\sqrt{n}},
$$
------------------------------------------------------------------------

where 
$$
f_n(x)=e^{\frac{x}{2}}\int^\infty_0 e^{-t}t^{n-\frac{1}{2}}\cos(2\sqrt{xt})dt,\quad x  \geq 0\;?
$$

**Proof** (by using Laguerre polynomials): it's easy to show that 
$$
f_n(x)=\sqrt \pi e^{-x/2} n! L^{(-1/2)}_n(x)
$$ 
and we know that 
$$
L^{(-1/2)}_n(x)=O\Big(e^{x/2}\frac{1}{\sqrt{n}}\Big).
$$
However, I'd like to not use this simple argument.