1. ## 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

2. 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.