Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail?

Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n \longrightarrow \mathbb{R}$ and suppose that $f$ is a positive convex function and we define $$ \varphi: \mathbb{R}^n \longrightarrow \mathbb{R}^n, \varphi(X) = \frac{\operatorname{grad}(f)(X)}{f(X)} $$

My question is this:

Is it true that the image of $\varphi$ is a convex set?

This is not my research area so I will appreciate any help or comment.