# Holomorphic function

• Jun 5th 2006, 02:24 AM
naty
Holomorphic function
What are all entire holomorphic function for which is Re f(z)<=Im f(z)
for every z from C?

Hint: use Picard theorem
• Aug 17th 2007, 02:01 AM
Rebesques
Since it is entire, we don't need to add "holomorphic". :)

Consider the function ${\rm e}^{f}.$ This is also entire and

$|{\rm e}^{f}|=|{\rm e}^{Ref+{\rm i}Imf}|={\rm e}^{Ref}\leq {\rm e}^{Imf}$, so the function ${\rm e}^{-Imf}{\rm e}^{f}={\rm e}^{f-Imf}$ is bounded. So it must be constant; That is, $f-Imf$ must be constant, or $(Ref-Imf)+{\rm i}Imf$ is constant. From this we get $Ref, Imf$ are constant.