Q:

Convexity of Gaussian Q-function (monotone decreasing)

I have a known convex function. If I take the Gaussian Q-function of this convex function would the resultant function also be convex?

A:

No, this is not even true when the convex function is $x$.
$1-\Phi(x)$ is not convex when $x < 0$. This can easily be seen by looking at the 2nd derivative of $1-\Phi(x)$, which is $\frac{x}{\sqrt{2\pi}} \text{exp}(-x^2/2)$.

