# Complex Analysis Problem

1. Sep 24, 2009

### Sistine

1. The problem statement, all variables and given/known data
Find all entire functions $$f$$ such that

$$|f(z)|\leq e^{\textrm{Re}(z)}\quad\forall z\in\mathbb{C}$$

2. Relevant equations
$$\textrm{Re}(u+iv)=u$$

3. The attempt at a solution

I tried using Nachbin's theorem for functions of exponential type. I also tried using the Cauchy integral formula to see if I could gain more information about $$f$$ but I could not solve the problem.

2. Sep 24, 2009

### Dick

I think you are looking at it in an overly complicated way. Can you think of a way to use |e^z|=e^(Re(z))?

3. Sep 25, 2009

### Sistine

I tried applying schwarz lemma to $$|f(z)|\leq |e^z|$$ i.e.

$$\left|\frac{f(z)}{e^z}\right|\leq 1$$

But this did not give me much information about $$f$$. What other Theorems from Complex Analysis could I use to gain information about $$f$$?

4. Sep 25, 2009

### Dick

Liouville's theorem!