Let $f:{\mathbb R}\rightarrow{\mathbb R}_+$ be a log-convex function. Suppose that $f_{\epsilon}$ is the smoothed version of $f$:

$$f_{\epsilon}(x)=\int \varphi_{\epsilon}(x-y)f(y)dy,$$

where $\varphi$ is a Mollifier. I would appreciate any pointers on relevant literature (or a hint) referring to conditions on $\varphi$ such that $f_{\epsilon}$ is log-convex.

Thanks.