Hi,

I don't know half of what the article is about, but I guess the estimate is because by definition of , and , the inequality being because we have a Banach algebra.

As for the second point, you can find the reference here (that's Hadamard's factorization theorem), and the definition of the order of an integral function is here. Here, because of the estimate, the order of is 1, and it has no zeroes, hence and is of degree 1. This is what you need.