I would like to ask a follow up on a [question I asked some days ago][1].

Consider the function 

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

My goal then was to analyze  

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

and Iosif Pinelis showed that this expression is constant.

Now instead of convolving with $f_0(x)=e^{-x^2}$ one could convolve with a function that decays slower than $e^{-x^2}$ for example $g_0(x)=e^{-\vert x \vert}.$ That is, we then get 

$$ G(y):=\frac{(f_2*g_0)(y)}{(f_0*g_0)(y)}- \left(\frac{(f_1*g_0)(y) }{(f_0*g_0)(y)}\right)^2.$$

Mathematica shows that $G$ is now not at all constant but has a unique maximum at $0$ and decreases from there. 

On the other hand one can consider the faster decaying function $h_0(x)=e^{-x^4}$ and consider

$$ H(y):=\frac{(f_2*h_0)(y)}{(f_0*h_0)(y)}- \left(\frac{(f_1*h_0)(y) }{(f_0*h_0)(y)}\right)^2.$$

In this case, the function has a unique minimum at zero and increases from there. 

All functions $F,G$ and $H$ are positive by the Cauchy-Schwarz inequality.

So to summarize for 

 - slow decay - $G$ has a unique minimum (Fig. 1)

[![$e^{-\vert x \vert}$][2]][2]


 - medium decay-$F$ is a constant function.
 - fast decay - $H$ has a unique maximum (Fig. 2).

 
[![$e^{-\vert x \vert^4}$][3]][3]

I also checked the exponent $e^{-\vert x \vert^{2+10^{-5}}}$ 

[![$e^{-\vert x \vert^{2+10^{-5}}}$][4]][4]

and it seems the function has the expected behaviour.

I also checked for log-convexity/log-concavity. Apparently, the functions $G,H$ are **neither** log-convex nor log-concave. 

See here a plot for $\log(G)$

[![log plot][5]][5]


  [1]: https://i.sstatic.net/2NvRL.jpg
  [2]: https://i.sstatic.net/cLSet.jpg
  [3]: https://i.sstatic.net/x06l1.jpg
  [4]: https://i.sstatic.net/xHvzC.jpg
  [5]: https://i.sstatic.net/GczB6.jpg