= Entire-function domination by another entire function
{title2=$|f|\leq C|g|\Longrightarrow f=cg$}
Let $f,g$ be <entire functions>, with $g$ not identically <zero>. If $|f(z)|\leq C|g(z)|$ everywhere, then $f/g$ is bounded and <holomorphic> away from the isolated <zeros> of $g$. The <removable singularity> extends it across each <zero>, giving a bounded <entire function>. <Liouville's theorem> makes it constant, so $f=cg$ with $|c|\leq C$. If $g$ is identically <zero> the inequality forces $f=0$. For example, domination by $|\cos z|$ forces a constant multiple of $\cos z$.
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