Math Help - Kahane, Zelazko paper. Problem with estimation.

1. Kahane, Zelazko paper. Problem with estimation.

Hi!
I'm studying Kahane, Zelazko paper "A characterization of maximal ideals in commutative Banach algebras" and I've problem with one statement in the proof of Theorem2.
Below is this theorem and part of the proof with 'red rectangle' - how can I proof/show this estimation and implication?
Any help will be highly appreciated.
Bes regards

[2] E. C. Titchmarsh, Theory of functions, Oxford 1939.

2. Hi,
I don't know half of what the article is about, but I guess the estimate is because $|f(x)|\leq\|f\|\|x\|$ by definition of $\|f\|$, and $\|e^x\|= \left\|\sum_{k=0}^\infty \frac{x^k}{k!}\right\|\leq \sum_{k=0}^\infty \frac{\|x\|^k}{k!}=e^{\|x\|}$, 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 $\lambda\mapsto e^{\psi(\lambda)}$ is 1, and it has no zeroes, hence $P=1$ and $Q$ is of degree 1. This is what you need.

3. Thanks for help. Now it is clear Once more thanks for quick answer.