Consider the function 

$$f_{n}(x)=e^{-x^2}x^n.$$

and the function 

$$h_p(x):=e^{-\vert x \vert^p}.$$

My goal is to analyze  

$$ F_p(y):=\frac{(f_2*h_p)(y)}{(f_0*h_p)(y)}- \left(\frac{(f_1*h_p)(y) }{(f_0*h_p)(y)}\right)^2$$

**Question:** Can we show that $F_p$ has a global maximum at zero for $p>2$ and a global minimum at zero for $p<2$?

Here are graphs from Mathematica for 

 - $F_1$, with a unique minimum

[![][2]][2]


 - $F_4$, with a unique maximum

 
[![][3]][3]

- $F_{2+10^{-4}}$, i.e. with an exponent slightly above $2$ 

[![][4]][4]

- $F_{2-10^{-4}}$, i.e. with an exponent slightly below $2$

[![][5]][5]

**Further observations:**

Iosif Pinelis showed that $F_2(y)$ is constant, see this earlier question of mine [question I asked some days ago][1].

All functions $F_p$ are positive by the Cauchy-Schwarz inequality.

The functions $F_p$ are not log-convex or log-concave in general. 

Finally, I computed the first derivative for convolving with $e^{-\vert x \vert^p}$ at $p=2$ by numerically differentiating 

[![][6]][6]

If you have any other conjectures you would like to me to verify, I am happy to do so. Just leave a comment!
  [1]: https://i.sstatic.net/2NvRL.jpg
  [2]: https://i.sstatic.net/VIFOi.jpg
  [3]: https://i.sstatic.net/H1N0U.jpg
  [4]: https://i.sstatic.net/4HMAP.jpg
  [5]: https://i.sstatic.net/Caurg.jpg
  [6]: https://i.sstatic.net/0uVdR.jpg