Considering the holomorphic function with its development in series at $0 $ : $f(z)=\sum_{n=0}^{+\infty}a_nz^n$
We suppose that f is converging withing $D=D(0,1)$ and that $∀z ∈ D, Re f (z) ≥ 0$
I have to show that $∀n ∈ N^∗,|a_n | ≤ 2Re(f(0))$.
What I did:
for $0<r<1$ and $n>0$, I proved that: $a_n=\frac{1}{\pi r^n }\int_{0}^{2\pi} \textit{Re}(f(re^{it}))e^{-int} dt $
But I have no clue on how to get $f(0) $ in the above formula and use for instance the Cauchy-Schwartz inequality to majorate $|a_n|$.
Any help, hints appreciated. Thank you