Entire function of exponential type $1$ bounded by $1/(1+|x|)$

Question

Let $f$ be an entire function of exponential type $1$ such that $|f(x)| \leq \frac{1}{1+|x|}$ for all $x\in \mathbb{R}$.

• First, I have to show that $|f(z)| \leq \frac{Ce^{|Im(z)|}}{1+|z|} z\in \mathbb{C}$.
• After that, I have to give an example of such a function (not $ 0$) and one more exemple of an entire function of exponential type 1 such that $|f(x)|=O(|x|^{-n}),x\in \mathbb{R},\forall n\geq 1$.

Progress

I have proved the first part of my question: We have to study the holomorphic function $g$ defined by $g(z)=f(z)e^{iz}(z+i)^{1/2}$. Now I am trying to find some examples.

Answer 1

As you did the first part, I do not address it. Concerning the examples, $c(\sin z)/z$ is the first example that you need. For the second example, take $\phi(t)$ infinitely differentiable, with support on $(-1,1)$, and consider the function $$f(z)=\int_{-\infty}^\infty\phi(t)e^{izt}dt.$$ This is evidently bounded: $|f(x)|\leq \|\phi\|_1$. Now, $$(-iz)^nf(z)=\int_{-\infty}^\infty\phi^{(n)}(t)e^{izt}dt,$$ (integration by parts), and we obtain that $|f(x)x^n|$ is bounded for all $n$.