Show that if $g$ and $h$ are holomorphic in $\mathbb{C}$ and $g\circ h$ is a no constant polynomial then $g$ and $h$ are polynomials.

Question

Let two entire functions(holomorphic in all $\mathbb{C}$) $g$ and $h$ so that the composition $g\circ h$ is a no constant polynomial. Show that $g$ and $h$ are polynomials.

Answer 1

If $g$ is not a polynomial, we know that $g$ takes a value say $c$ an infinity of time, ie there exists an infinite sequence $z_n$ of distinct points such that $g(z_n)=c$. As $h$ is entire, we know that at most one value of the sequence $z_n$ is not in the range of $h$, ie we can find a sequence $y_n$ of different points such that $h(y_n)=z_n$ for all but one index. So we can find an infinite sequence of distinct points $y_n$ such that $g\circ h (y_n)=c$, and $g\circ h$ is not a polynomial.

So $g$ is a polynomial.

It is even simpler to prove that $h$ is a polynomial : if not there exists a infinite sequence of distinct complex numbers $z_n$ such that $h(y_n)= c$, then $g\circ h (z_n)=c$ and this function is not a polynomial.