Let $f$ be an entire function with $f(3^{-n})=3^{-(n+1)}\: \forall\: n\geq 1$.
I am trying to determine $f(z)$ for all $z$.
I have established that $f$ has an isolated zero at $z=0$ of order 1, and that $f$ is not bounded (Liouville's Theorem). After sifting through all of the theorems presented in lectures that relate to this topic, I cannot see how I can fully identify $f$. Also that $f(\mathbb{C})=\mathbb{C}$ or $\mathbb{C}$ minus one point. (Picard's Little Theorem).
I have just covered the Identity Theorem, Rouche's Theorem, open mapping theorem, schwarz' lemma, maximum principle, Picard's little theorem etc. but I am still stumped.
Any hints to point me in the right direction would be much appreciated.
See $f(z)-\frac z3$ has an accumulate point of zeros (i.e. $3^{-n}\to 0$), so it must be identically $0$, i.e. $f(z)=\frac z3$.