Entire Functions Bounded by Exponential Growth

[Sistine]

Homework Statement

Find all entire functions $$f$$ such that

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

Homework Equations

$$\textrm{Re}(u+iv)=u$$

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.

 

[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))?

 

[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 $$f$$?

 

[Dick]

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 $$f$$?

Liouville's theorem!