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 function $e^{-\vert x \vert^{2+10^{-5}}}$, i.e. with an exponent slightly above $2$ [![$e^{-\vert x \vert^{2+10^{-5}}}$][4]][4] and it seems the function has the expected behaviour, i.e. there exists a unique maximum at zero and the function decays from there. One sees the opposite effect by plotting **Addendum:** 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] And finally, I computed the first derivative for convolving with $e^{-\vert x \vert^p}$ at $p=2$ by numerically differentiating (so be careful about the result please [![first derivative][6]][6] [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 [6]: https://i.sstatic.net/Rfhsn.jpg