Composition of non constant entire functions is non constant

Question

Let $f,g$ be non constant entire functions. I have to prove that the compositions is not constant.

I have thought about using the derivative of the composition:

$(f o g)'(z) = f'(g(z))g'(z)$

and use the fact the both derivatives are cero only in isolated points, so the composition is not constant but I don't know if it is sufficient.

Thanks in advance.

Answer 1

Your method works.

You can also say that $f(\Bbb C)$ is an non-empty open set, by the open mapping theorem and because $f$ is not constant. Therefore, by the identity theorem, and because $g$ is not constant, the restriction of $g$ to $f(\Bbb C)$ is not constant.

Answer 2

Assume that the composition $f\circ g$ is constant. Then $|f(g(z))|=M$ for some $ M>0$. Now, as $g$ is non- constant entire which implies that Range of $g$ contains an open disc as range of an entire functions skips atmost one point. Then $|f|=M$ is constant in that disc, hence, by identity theorem, $f$ is bounded which makes $f$ constant by Louville's theorem.