Characterize all analytic functions satisfyinf the given condition > Characterize all analytic functions $f(z)$ in $|z|<1$ such that $|f(z)|\le |\sin(1/z)|$ , for all $0<|z|<1$. I can't understand from where I will start ?

First prove that a non-constant analytic function can have at most finitely many zeros in a bounded, compact set. Proof idea: if it's bounded and compact, and has infinitely many zeros, there must be an accumulation point, so the function must be identically zero.

Now notice that $z_i=1/(n\pi)$ is such a sequence for $f(z)$.