# Thread: Louisville's Theorem?

1. ## Louisville's Theorem?

The function f is holomorphic on $\mathbb{C}$ and it is given that there exists a K > 0 such that

$|f(z) + f(-z)| \le K$ for all z.

What can we say about f?

I was thinking perhaps Louisville's theorem would explain this, (ie bounded entire functions are constant) but I'm not sure how to prove this. Or perhaps f is an odd function. Any help with this would be appreciated

2. Originally Posted by slevvio
The function f is holomorphic on $\mathbb{C}$ and it is given that there exists a K > 0 such that

$|f(z) + f(-z)| \le K$ for all z.

What can we say about f?

I was thinking perhaps Louisville's theorem would explain this, (ie bounded entire functions are constant) but I'm not sure how to prove this. Or perhaps f is an odd function. Any help with this would be appreciated

For any function we have $f(z)=\frac{f(z)+f(-z)}{2}+\frac{f(z)-f(-z)}{2}$ (sum of even plus odd functions) ...

Tonio

3. Originally Posted by slevvio
The function f is holomorphic on $\mathbb{C}$ and it is given that there exists a K > 0 such that

$|f(z) + f(-z)| \le K$ for all z.

What can we say about f?

I was thinking perhaps Louisville's theorem would explain this, (ie bounded entire functions are constant) but I'm not sure how to prove this. Or perhaps f is an odd function. Any help with this would be appreciated
On the basis of Liousville theorem You can derive that the even part of $f(*)$ is a constant or alternatively that $f(*)$ is the sum of a constant and an odd function...

Kind regards

$\chi$ $\sigma$