Entire functions that satisfy certain equation. Problem
Find all the entire functions that satisfy $$n^2f(\dfrac{1}{n})^3+f(\dfrac{1}{n})=0 \space \space \forall \space n \in \mathbb  N$$
My idea was to define a function $$g(z)=z^2f(z)^3+f(z)$$
Since $f$ is an entire function and $g$ is composition of entire functions, then $g$ is entire as well. The function $g$ vanishes on the sequence $\{\dfrac{1}{n}\}_{n \in \mathbb N}$ and $\dfrac{1}{n} \to 0 \in \mathbb C$, then $g$ is identically zero (if a function $f$ defined on a region $\Omega$ vanishes on a sequence of different points with a limit point in $\Omega$, then $f$ is identically zero).
We have $$0=g=z^2f(z) ^3+f(z),$$ from here we obtain $$f(z)=-z^2f(z)^3$$
I have the feeling that I have not answered the complete solution, but I don't know what else I can say about $f$. I would appreciate some help to complete my solution or suggestions of another answer.
 A: To make your original attempt work,, rewrite the given condition as 
$$f(\frac 1n)^3+ \frac1{n^2}f(\frac1n)=0$$
and let $g(z)=f(z)^3+z^2f(z)$. As before, $g(\frac1n)=0$ and hence $g(z)=0$.
Then $$0=f(z)+z^2f(z)= f(z)(f(z)-iz)(f(z)+iz).$$
At least one factor must be $=0$ for a converging sequence, hence is identically zero. We conclude that $f(z)=0$, $f(z)=iz$, $f(z)=-iz$ are the only three solutions.
A: Clearly $f(z)=0$ is a solution, so assume not. Then the functional equation is:
$$z^{-2}f(z)^3+f(z)=0$$
satisfied on some infinite subsequence of $\{{1\over n}:n\in\mathbb{N}\}$ where $f(z)\ne 0$, which is possible since $f\not\equiv 0$.
As a preliminary note, this is an analytic function because the original functional equation implies $f(0)=0$ so that the Taylor series for $f$ has lowest term at least degree 1, so when we cube it the lowest power is at least 3, so the pole ostensibly introduced by the $z^{-2}$ term is seen to be removed, so that the function is analytic.
Then we see that $f(z)^2=-z^2$ along this subsequence, which implies $f(z)=\pm iz$ since that is a set with an accumulation point in the domain $\mathbb{C}$.
