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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 f {\displaystyle f} and g {\displaystyle g} are entire, and | f | ≤ | g | {\displaystyle |f|\leq |g|} everywhere, then f = α g {\displaystyle f=\alpha g} for some complex number α {\displaystyle \alpha }. Consider that for g = 0 {\displaystyle g=0} the theorem is trivial so we assume g ≠ 0 {\displaystyle g\neq 0}. Consider the function h = f / g {\displaystyle h=f/g}. It is enough to prove that h {\displaystyle h} can be extended to an entire function, in which case the result follows by Liouville's theorem. The holomorphy of h {\displaystyle h} is clear except at points in g − 1 (0) {\displaystyle g^{-1}(0)}. But since h {\displaystyle h} is bounded and all the zeroes of g {\displaystyle g} are isolated, any singularities must be removable. Thus h {\displaystyle h} can be extended to an entire bounded function which by Liouville's theorem implies it is constant.