I am currently working on some review problems in complex analysis and came upon the following conundrum of a problem.
"If $f(z)$ is an entire function, and satisfies $|f(z^2)|\le|f(z)|^2$, prove that f(z) is a polynomial."
My intuition tells me to show that f(z) has a pole at infinity by showing that infinity is not an essential or removable singularity. However, I am getting stuck after this.
Thanks for the help,