Characterize an analytic function with restriction of its growth

Question

Characterize all analytic functions $f(x)$ in $|z|<1$ such that $|f(z)|\leq|\sin(1/z)|$ for all points in punctured disk.

I think we should change the form of $\sin(1/z)$ to find a connection with polynomial which $f(x)$ can be expanded into. But I don't know how to do that.

Answer 1

Hint:

Take the sequence $\{z_n\}=\left\{\frac{1}{n\pi}\right\}$ and use Identity theorem.

What is $f(z_n)$ ?