Prove $f(z)=-\frac{\log(1-z)}{z}$ is convex in $\mathbb{D}$ 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.

 A: Here is a proof which makes use of Maple, namely, its DirectSearch package created by Sergey Moiseev. The GlobalOptima command finds the global minimum of
$${\Re}\left(1+z\dfrac{f''(z)}{f'(z)}\right)$$ in the unit disk with the absolute error less than or equal to $10^{-12}$:
f := -log(1-z)/z:
A := evalc(Re(eval(1+z*(diff(f, z, z))/(diff(f, z)), z = x+I*y))):
DirectSearch:-GlobalOptima(A, {x^2+y^2 <= 1}, tolerances = 10^(-12));


[ 0.294349724781047,[x=- 0.999999999999916,y= 0.0000004101682696],498] 

B := evalc(Re(eval(1+z*(diff(f, z, z))/(diff(f, z)), z = x+I*sqrt(-x^2+1)))):
DirectSearch:-GlobalOptima(B, {x >= -1, x <= 1}, tolerances = 10^(-12));


[ 0.294349724781044,[x=- 0.999999999999999],126]

These outputs mean that the global minimum equals $0.2943497247810$. I am pretty sure that the results imply the affirmative answer to the question. However, colleagues are welcome to discuss it.
Addition. Just for illustration, here is a plot of A on the unit disk.
plot3d(A, y = -sqrt(-x^2+1) .. sqrt(-x^2+1), x = -1 .. 1, axes = frame);


In fact, the function is unbounded from above around $z=1$.
A: The normalization of f, (f-1)/2 is the image of the fundamental convex function z/(1-z) under the operator H(g) = (2/z)* primitive g.
It is a theorem of Libera (Goodman, Univalent Functions 1983, vol 2, page 156) that while H doesn't preserve the S class (schlicht), it does preserve the classes CV (convex S), ST (starlike S), CC (close to convex S), so the answer to your question follows.
The proof is based on a lemma about p-valent starlike functions (here we need the 2-valent primitive of g, for g in one the S classes above) and a result giving a (necessary) and sufficient condition for a function to belong to P (functions on the unit disk with ReP > 0 and P(0)=1, so it is not obvious
A: http://www.ams.org/journals/proc/1965-016-04/home.html
The original article of Libera1 is in Proceedings AMS, 16, 1965, p 755-758. freely available at the link above
The proof there is pretty much the one I read in the Goodman book 
1R. J. Libera. Some classes of regular univalent functions,
Proc. Amer. Math. Soc. 16 (1965), 755-758; doi: 10.1090/S0002-9939-1965-0178131-2
