I am playing a bit with the Weierstrass factorization of entire functions and I have a question on bounds.
Let $a_k$'s be such that $\sum_{k\ge0}a_k^{-1}$ converges and let $$ f(z) = \prod_{k\ge0}\left(1 - \frac{z}{a_k} \right) .$$
What is the order of $f$? Is it majorated?
Similarly if $\sum_{k\ge0}b_k^{-2}$ and $$ g(z) = \prod_{k\ge0}\left(1 - \frac{z^2}{b_k^2} \right) ,$$ what is the order of $g$? Is it majorated?
Is there a standard way to deal with this kind of questions?