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

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

• $F_4$, with a unique maximum

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

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

Further observations:

Iosif Pinelis showed that $F_2(y)$ is constant.

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