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## Amazing property of entire functions

Prove that if $p,q,r$ are non-constant, non-vanishing entire functions with $p+q+r=0$ then there exists an entire function $h$ such that $p,q,r$ are constant multiples of $h$.

Solution
Because each of the given function do not vanish, we can write $\frac{p}{r}+\frac{q}{r}=-1$. Now look at the function $\frac{p}{r}$. This is an entire function, as a quotient of two non-vanishing holomorphic functions. Because $\frac{q}{r} \neq 0$ also, we find that $\frac{p}{r}$ avoids two complex numbers, namely $0$ and $-1$. A weak variant of Picard’s Theorem says that if this happens, then $\frac{p}{r}$ is constant, and so is $\frac{q}{r}$. Therefore, $p,q,r$ are scalar multiples of $r$.