Let $k$ be an integer with $k>1$ and let $$a\_k=\frac{2 k(k-1)}{\Gamma[k-1]} \int\_0^\infty \frac{e^{-r^2} r^{2k-3}}{2+4 r^2+r^4} dr.$$


How to prove that $a\_k<1$?



A bounded sequence $a_k=\frac{2 k(k-1)}{\Gamma[k-1]} \int_0^\infty \frac{e^{-r^2} r^{2k-3}}{2+4 r^2+r^4} dr$
$\newcommand\Ga\Gamma$We have
$$a\_k=\frac{2 k(k-1)}{\Ga(k-1)} J\_k,\quad J\_k:=\int\_0^\infty e^{-r^2} r^{2k-3}g(r)\,dr,$$
$$g(r):=\frac1{2+4 r^2+r^4}.$$
Next (for $r>0$),
$$g(r)=h(r)-\frac{8 \left(70 r^2+41\right)}{r^{12} \left(r^4+4 r^2+2\right)}<h(r),$$
where
$$h(r):=\frac{1}{r^4}-\frac{4}{r^6}+\frac{14}{r^8}-\frac{48}{r^{10}}+\frac{164}{r^{12}}.$$
So, for $k\ge8$
$$a\_k<\frac{2 k(k-1)}{\Ga(k-1)}\,\int\_0^\infty e^{-r^2} r^{2k-3}h(r)\,dr
=\frac{(k-1) k (k (k ((k-26) k+265)-1296)+2768)}{(k-7) (k-6) (k-5) (k-4) (k-3) (k-2)}<1$$
if $k\ge14$.




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It remains to check the inequality $a\_k<1$ for $k=2,\dots,13$. This is straightforward to do. For instance,
$$a\_2=\frac{e^{2-\sqrt{2}} \left(e^{2 \sqrt{2}}
 \text{Ei}\left(-2-\sqrt{2}\right)-\text{Ei}\left(-2+\sqrt{2}\right)\right)}{\sqrt{2}}
=0.427\ldots<1.$$
Mathematica can compute values of the built-in special function $\text{Ei}$ with any degree of accuracy.


More generally, one can note that
$$K\_2(t):=\int\_0^\infty e^{-t\,r^2} r^{2\times2-3}g(r)\,dr
=-\frac{e^{-\left(\left(\sqrt{2}-2\right) t\right)} \left(\text{Ei}\left(\left(-2+\sqrt{2}\right) t\right)-e^{2 \sqrt{2} t}
 \text{Ei}\left(-\left(\left(2+\sqrt{2}\right) t\right)\right)\right)}{4 \sqrt{2}},$$
which follows by the partial fraction decomposition
$$\frac{2 \sqrt{2}}{2+4 u+u^2}=-\frac{1}{u+\sqrt{2}+2}-\frac{1}{-u+\sqrt{2}-2}.$$
Then one can use the formulas
$$J\_k=(-1)^{k-2}K\_2^{(k-2)}(1)$$
for $k\ge2$ and $\text{Ei}'(u)=e^u/u$ to find $J\_k$ in closed form for $k=2,\dots,13$, to get
$$(a\_2,\dots,a\_{13})=(0.42\ldots,0.66\ldots, 0.78\ldots, 0.85\ldots, 0.89\ldots, 0.91\ldots, \\ 
0.93\ldots, 0.95\ldots, 
0.95\ldots, 0.96\ldots, 0.97\ldots, 0.97\ldots).$$
In particular,
$$a\_{13}=\frac{13}{1663200}\\ 
\times\left(-8 \left(8119+5741 \sqrt{2}\right) e^{2+\sqrt{2}} \text{Ei}\left(-2-\sqrt{2}\right)+8 \left(5741 \sqrt{2}-8119\right) e^{2-\sqrt{2}}
 \text{Ei}\left(-2+\sqrt{2}\right)+94224\right)
=0.97\ldots.$$


