Characterize all entire functions $f$ such that $|f(z)|=1$, whenever $|z|=1$

Question

Characterize all entire functions $f$ such that $|f(z)|=1$, whenever $|z|=1$.

I think either $f(z)=c$ or $f(z)=cz$ where $|c|=1$. But I am not sure, any hint would be great, thanks.