I am currently trying to solve the following problem from a previous qualifying exam.
Let $\alpha,\beta: \mathbb{C}\to \mathbb{C}$ be two non-constant entire functions with exactly the same zeroes of the same order. Assume that, for all $z\in \mathbb{C}$, $|\alpha(z)|\leq (1+|z|)^5|\beta(z)|$ and $\alpha(0)=1, \beta(0)=2$. Express $\beta$ in terms of $\alpha$.
I know that if an analytic function $f$ has a zero at $z=a$, then we can write $f(z)=(z-a)^mf_1(z)$, where $f_1$ is analytic and $f_1(a)\neq 0$. But I am not sure how to proceed further. Any help/hint will be very useful. Thanks in advance.