Let $p, q, r$ be non-vanishing non-constant entire holomorphic functions such that $$p+q+r = 0$$ Does there exist an entire function $h$ such that $p, q, r$ are the constant multiplies of $h$?
2026-03-25 06:06:45.1774418805
Does there exist an entire function $h$ such that $p, q, r$ are the constant multiplies of $h$?
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Since $r$ doesn't vanish, then $p/r$ and $q/r$ are entire and don't vanish.
We have that $p/r+q/r=-1$. By Picard's theorem either $q/r$ is constant or there is a point in which $q/r=-1$, since $q/r$ already skips the value $0$ and it can't skip yet another one. At that point $p/r$ would vanish, which is a contradiction. Therefore $q/r$ is constant and therefore so is $p/r$. Take $h:=r$. Then $p=(p/r)h$, $p=(q/r)h$, and $r=1h$.