What are all entire holomorphic function for which is Re f(z)<=Im f(z)
for every z from C?
Hint: use Picard theorem
Since it is entire, we don't need to add "holomorphic".
Consider the function $\displaystyle {\rm e}^{f}.$ This is also entire and
$\displaystyle |{\rm e}^{f}|=|{\rm e}^{Ref+{\rm i}Imf}|={\rm e}^{Ref}\leq {\rm e}^{Imf}$, so the function $\displaystyle {\rm e}^{-Imf}{\rm e}^{f}={\rm e}^{f-Imf}$ is bounded. So it must be constant; That is, $\displaystyle f-Imf$ must be constant, or $\displaystyle (Ref-Imf)+{\rm i}Imf$ is constant. From this we get $\displaystyle Ref, Imf$ are constant.