  Let $X$ be an arbitrary continuous random variable and $Z$ be an independent
Gaussian random variable with zero mean and unit variance. For $t~>~0$, Costa
proved that $e^{2h(X+\sqrt{t}Z)}$ is concave in $t$, where the proof hinged on
the first and second order derivatives of $h(X+\sqrt{t}Z)$. Specifically, these
two derivatives are signed, i.e., $\frac{\partial}{\partial t}h(X+\sqrt{t}Z)
\geq 0$ and $\frac{\partial^2}{\partial t^2}h(X+\sqrt{t}Z) \leq 0$. In this
paper, we show that the third order derivative of $h(X+\sqrt{t}Z)$ is
nonnegative, which implies that the Fisher information $J(X+\sqrt{t}Z)$ is
convex in $t$. We further show that the fourth order derivative of
$h(X+\sqrt{t}Z)$ is nonpositive. Following the first four derivatives, we make
two conjectures on $h(X+\sqrt{t}Z)$: the first is that
$\frac{\partial^n}{\partial t^n} h(X+\sqrt{t}Z)$ is nonnegative in $t$ if $n$
is odd, and nonpositive otherwise; the second is that $\log J(X+\sqrt{t}Z)$ is
convex in $t$. The first conjecture can be rephrased in the context of
completely monotone functions: $J(X+\sqrt{t}Z)$ is completely monotone in $t$.
The history of the first conjecture may date back to a problem in mathematical
physics studied by McKean in 1966. Apart from these results, we provide a
geometrical interpretation to the covariance-preserving transformation and
study the concavity of $h(\sqrt{t}X+\sqrt{1-t}Z)$, revealing its connection
with Costa's EPI.
