One to one map $f$ equal to its power series Across a difficult exercise sheet I encountered this exercise : 

Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one (injective) and that $f(z)= \sum_{n \geqslant 0} a_n z^n$ in $D$ (the unit open disk) where $\sum a_n z^n$ has a convergence radius equal to $1$ Show that $\sum a_n z^n$ does converge uniformly on $\bar D$

The proof is hard and involves a bit of complex analysis, and some really tricky results of Fourier analysis. 
My question is the following : what is the name of this theorem and where can I find it "in context" ? Moreover, my knoledge is just undergraduate Fourier analysis, so is there a reference book for Fourier analysis (especially for Fourier series) and power series results that are not well known ? In fact I think that this theorem is not well known ^^ 
Thank you. 
EDIT. If someone is interested in I can provide a detailed proof of this problem, I would be glad to do it if someone ask for. 
 A: Well, someone asked for so there is a compact solution (I can provide more details of course) :
First, since $f$ is one to one we deduce that $f'(z) \neq 0$ for all $z \in \bar D$. Then we set $D_r = \{ z, \; |z| \leqslant r\}$. Then if $A_r$ denotes the area of $f(D_r)$ we have : $$A_r= \iint_{f(D_r)} \mathrm{d}x \, \mathrm{d}y =\iint_{D_r} |f'(x+iy)|^2 \mathrm{d}x \, \mathrm{d}y = \int_{0}^{r} I(\rho) \mathrm{d} \rho$$
where $ \displaystyle I (\rho) = \int_{0}^{2\pi} |f'(\rho e^{i \theta})|^2\mathrm{d} \theta = 2 \pi \sum_{n \geqslant 1} (na_n \rho ^n)^2$ using Parseval-Plancherel identity.
A classical sum/integral inversion shows that : $$A_r = 2 \pi \sum_{n \geqslant 1} n|a_n|^2 r^{2n}$$
for any $r<1$. Thus for any $N,r$ we have : $$\sum_{k=1}^{N} n|a_n|^2 r^{2n} \leqslant A_1$$ Since the bound is uniform in $r$ and $N$ the convergence of $\sum n|a_n|^2$ is ensured. 
Next we set $\displaystyle S(N)= \sum_{n=1}^N n|a_n|$ we can write using Cauchy-Schwarz : $$S(N) \leqslant \sqrt{\sum_{k \leqslant N} k} \sqrt{\sum_{k \leqslant N} n |a_n|^2} = \mathcal{O} (N) $$
This is the key estimate [Q1]
Let $g(\theta) = f(e^{i \theta})$ and $g_{r} (\theta) = f(re^{i \theta})$. By the dominated convergence theorem we can easily show that $\hat g (n) =a_n$. 
Then if $F_N$ is the Fourier kernel we can set $\phi _N (\theta) = F_N * g (\theta) = \sum_{n=0}^N \left( 1- \frac{n}{N+1} \right) a_n e^{in \theta}$. We know that $\phi _N \to g$ uniformly on $\mathbb{R}$. [Q2] Plus we write $\phi _N = S_N - \psi _N $ so that : $$|\psi _N (\theta) | \leqslant \frac{\sum_{n=0}^N n|a_n|}{N+1} \to 0$$ We conclude that $S_N$ converges uniformly on the circle so in the entire disk (Abel transform inside). [Q3]
Thanks to this short version of the proof I can ask a few questions : 
[Q1] Where can I find "sharp" estimates like this one, on the Fourier coefficients. Because all I know is the Fourier Plancherel identity and the Cauchy integral representation of the Fourier coefficients. But I am convinced a lot of work in the direction of more accurate estimations have been done. 
[Q2] All I know about kernels in Fourier analysis is Fejer, Dirichlet and Poisson's one. Is there a reference in this direction ? Because we see here how important the kernel regularisation is important ! 
[Q3] This fact is well knonw and often proposed as an exercise (it was the case for me). But is there a lot of propreties like that about power series ? 
