Let $f:{\mathbb R}\rightarrow{\mathbb R}\_+$ be a [log-convex](http://en.wikipedia.org/wiki/Logarithmically_convex_function) 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](http://en.wikipedia.org/wiki/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.



log-convexity of Mollified function?
Yes if you assume $\phi\_\epsilon\ge0$. Because $f\_\epsilon$ belongs to the convex cone spanned by the shifted functions $f(\cdot-h)$. Since every $f(\cdot-h)$ is logarithmically convex and the set of logarithmically convex functions is itself a convex cone, you are done.


