Suppose that $f$ is holomorphic on the unit disk $|z|<1$. If $\,\exists$ $r \in (0,1)$ such that $|f(1/n)|\leq r^n$ for $n \in \mathbb{N}$. Then $f=c$ (constant).
I think this problem could be solved by using Cauchy's inequalities formula ( I am not sure) which is: If $f$ is holomorphic in an open set $\Omega$ that contains the closure $C$ of a disc $D$, centered at $z$ and has radius $R$, then
$|f^{(n)}(z)|\leq \dfrac{n! ||f||_C}{R^n} $,
where $||f||_C=\sup_{z \in C}|f(z)|$.
But I don't know how to use it. What is my disc (What are its radius and center?) I have exam in two days, any help is appreciated.