Suppose that $f$ is analytic in the unit disc D = {$z \in \mathbb{C}$ : |$z$| < 1} and $|$f($z$)$| \le 1/(1-|$z$|)$ for all $z\in D$.
Let $f($z$)= \sum _{n=0}^{\infty } a_nz^n$ be the power series expansion of f about $0$.
Prove that $|a_n| \le (n+1)(1+1/n)^{n} < e(n+1)$