Complex Analysis Problem

  • Thread starter Sistine
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  • #1
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Homework Statement


Find all entire functions [tex]f[/tex] such that

[tex] |f(z)|\leq e^{\textrm{Re}(z)}\quad\forall z\in\mathbb{C}[/tex]


Homework Equations


[tex]\textrm{Re}(u+iv)=u[/tex]


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 [tex]f[/tex] but I could not solve the problem.
 

Answers and Replies

  • #2
Dick
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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
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I tried applying schwarz lemma to [tex]|f(z)|\leq |e^z|[/tex] i.e.

[tex]\left|\frac{f(z)}{e^z}\right|\leq 1[/tex]

But this did not give me much information about [tex]f[/tex]. What other Theorems from Complex Analysis could I use to gain information about [tex]f[/tex]?
 
  • #4
Dick
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I tried applying schwarz lemma to [tex]|f(z)|\leq |e^z|[/tex] i.e.

[tex]\left|\frac{f(z)}{e^z}\right|\leq 1[/tex]

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

Liouville's theorem!
 

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