I am interested in understanding the properties of entire functions of exponential type 1. I have few questions about their growth.
How many sectors can a function of exponential type have, in which its modulus increases exponentially? Do zeros have to stay away from such sectors?
If we restrict the exponential type to 1, does the answer to the above question change?
I was looking into the Weierstrass factorization of entire functions from which we get (assuming that I didn't misunderstand anything) that if $f$ is of exponential type and $A=\{a_1,a_2,\dots\}$ is the set of its zeros, then we have 2 cases:
either $$ f(z) = M z^m e^{b z} \prod_{a_i\in A} \Big(1-\tfrac{z}{a_i}\Big) e^{\frac{z}{a_i}} $$ for some $M$, $b\in\mathbb{C}$ and $ \sum_{a_i\in A} a_i^{-2} <\infty$,
or $$ f(z) = M z^m \prod_{a_i\in A} \Big(1-\tfrac{z}{a_i}\Big) $$ for some $M\in\mathbb{C}$ and $ \sum_{a_i\in A} a_i^{-1} <\infty$.
Am I correct with this? Are both cases valid?
What can we say more if we further assume that the exponential type is 1? Does it mean that $|b|\le 1$, for example?
How can one majorate products of the form $$ \prod_{a_i\in A} \Big(1-\tfrac{z^2}{a_i^2}\Big) $$ with $ \sum_{a_i\in A} a_i^{-2} <\infty$, or products of the form $$ \prod_{a_i\in A} \Big(1-\tfrac{z}{a_i}\Big) $$ with $ \sum_{a_i\in A} a_i^{-2} <\infty$? Is there a standard machinery for that? If they are infinite, what kind of entire functions these products define? Are there of order 1? If yes, what can be said about their type?
