Certain condition on an entire function implies the function is constant ? Let $f=u+iv$ be an entire function such that $v \ge 2u+1$ , then is it true that $f$ is constant ? 

Just to throw another (almost equivalent) method in, $0$ is not in the image of the function and there is $r>0$ such that the disc $D_r=\{z\ :\ |z|<r\}$ is not either.
For example, take $r=1/4$: if $|z|<1/4$, then, $z=x+iy$ with $|x|,|y|<1/4$, so
$3/4\geq|y-2v|$, hence it is impossible that $y\geq 2x+1$ when $x+iy=z\in D_{1/4}$.
Now, take $g=1/f$. This is a holomorphic entire function. We know that $|f|>1/4$, therefore $|g|<4$, so by Liouville's theorem, $g$ is constant and so is $f$.
