Entire-function domination by another entire function ($|f|\le C|g|⟹f=cg$)

Let $f,g$ be entire functions, with $g$ not identically zero. If $|f(z)|\le C|g(z)|$ 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 $|c|\le C$. If $g$ is identically zero the inequality forces $f=0$. For example, domination by $|\cos z|$ forces a constant multiple of $\cos z$.