Let $\alpha$ be a real number and $C$ a positive number. Determine all entire functions $f$ such that $$|f(z)|\le C(1+|z|)^\alpha \ \ \ \ \ \ \ \ \ \ \forall z\in\mathbb{C}$$
I don't recall seeing anything like this throughout our entire course, and my textbook is completely devoid of something that looks like this. So I'm at a total loss.
I imagine it may include polynomials, since we'd need to have: $$\left|\frac{f(z)}{(1+|z|)^\alpha}\right|<C$$ so for any particular polynomial, $p(x)=a_0+...+a_nz^n$ we can choose $\alpha>n$, But I'm not sure about how to determine ALL entire functions.