Problem 1612 (difficulty: 5/10)

Let \(\displaystyle \displaystyle f(z)=\sum_{n=0}^\infty a_nz^n\) be entire that satisfies \(\displaystyle |f(z)|<e^{|z|}\). Prove that \(\displaystyle |a_n|\le\left(\frac{e}{n}\right)^n\).

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