find all entire functions that are 1-1 That is all the given information.
I should find all the entire functions that are one to one, I have no clue on how to do that.
And there is also another one, that goes like this:
Find all entire functions that are:
$$\forall z \in \mathbb C^* \;\;\;\;\; \lvert f(z)\rvert \le \lvert \frac{\sin z}{z} \rvert$$
$$\mathbb{C}^* \text{ is complex plane without } (0,0)$$

For the first one, there are four cases to consider: constant functions, polynomials of degree $1$, polynomials of degree $>1$, and non-polynomials. The first two cases are obvious. For the third case the derivative is a non-constant polynomial and thus has zeroes, and holomorphic functions are not one-to-one near where their derivatives vanishes. For the last case, $f(\frac{1}{z})$ has an essential singularity and thus takes on almost every value in $\mathbb{C}$ repeatedly as you go to the origin (Picard's theorem). So only the degree $1$ polynomials are injective.
