Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be an entire function such that $|f(1/n)|\leq 1/n^{3/2}$ for all $n\in \mathbb{N}$ then show that $\{n^2f(1/n)\}$ is bounded.
To show that $\{n^2f(1/n)\}$ is bounded, we need $M>0$ such that $|n^2f(1/n)|\leq M$ equivalently $|f(1/n)|\leq M/n^2$.
We have $|f(1/n)|\leq 1/n^{3/2}$ and we want to prove $|f(1/n)|\leq M/n^2$..
I could not think of any method..
Suggest some hints..
EDIT : Suppose it is unbounded, given $M>0$ there exists $n\in \mathbb{N}$ such that $|n^2 f(1/n)|>M$ in particular, for each $i\in \mathbb{N}$ we have $n_i\in \mathbb{N}$ such that $|f(1/n_i)|\geq i/n_i^2$..
I do not see what property of analytic function i have to use in this case..
Hint: $f$ has a zero at $0$. What can you say about its order?