Editing Liouville's theorem (complex analysis) (section)

===No entire function dominates another entire function===
A consequence of the theorem is that "genuinely different" entire functions cannot dominate each other, i.e. if <math>f</math> and <math>g</math> are entire, and <math>|f|\leq |g|</math> everywhere, then <math>f=\alpha g</math> for some complex number <math>\alpha</math>. Consider that for <math>g=0</math> the theorem is trivial so we assume <math>g\neq 0</math>. Consider the function <math>h=f/g</math>. It is enough to prove that <math>h</math> can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of <math>h</math> is clear except at points in <math>g^{-1}(0)</math>. But since <math>h</math> is bounded and all the zeroes of <math>g</math> are isolated, any singularities must be removable. Thus <math>h</math> can be extended to an entire bounded function which by Liouville's theorem implies it is constant.