For analytic function $f:\mathbb{C}\to\mathbb{C}$ with

$$f(z)=-\dfrac{\log(1-z)}{z}$$

I want to prove that it's a convex map in the unit disk $\mathbb{D}$, i.e. that $f$ maps the unit disk conformally onto a convex domain. I know $${\bf Re}\left(1+z\dfrac{f''(z)}{f'(z)}\right)>0$$ for $z\in\mathbb{D}$, if and only if function be convex in $\mathbb{D}$ [1], but I couldn't use it in my problem. Is the any way to prove that $f$ is convex in $\mathbb{D}$.

Thank.

- Duren Peter L., Univalent functions (Grundlehren der mathematischen Wissenschaften 259), Springer-Verlag Berlin and Heidelberg GmbH & Co. K (1983), p.42.