Problem with holomorphic function and series

Question

Lets $f:D(0,1)\rightarrow C$ holomorphic with $|f(z)|\leq\frac{1}{1-|z|}$ , $\forall z\in D(0,1)$. If $f(z)= \sum_{i=0}^nc_nz^n$ prove that $|c_n| \leq(n+1)\big(1+\frac{1}{n}\big)^n$ $n\in N$

Answer 1

We know $c_n=\dfrac{f^{(n)}(0)}{n!}$ then according to Cauchy integral formula $$c_n=\dfrac{1}{2\pi i}\int_{|z|=r}\dfrac{f(z)}{z^{n+1}}dz$$ and for $r<1$ $$|c_n| \leqslant \dfrac{1}{2\pi r^n}\int_0^{2\pi}|f(re^{i\theta})|d\theta \leqslant \dfrac{1}{2\pi r^n}\int_0^{2\pi}\dfrac{1}{1-r}d\theta = \dfrac{1}{r^n(1-r)}$$ for $r=\dfrac{n}{n+1}$ the function $\phi(r)=\dfrac{1}{r^n(1-r)}$ is maximum and with substitution you find $|c_n| \leqslant (n+1)\left(1+\dfrac1n\right)^n$.