A problem on Maximum-modulus principle

Question

I was reading a proof of a problem. I encountered the following which I can't understand: $D$ denotes the open unit disc

$h:D\to D$ defined as $h(z)=a_{1}z+a_{3}z^{3}+....$. It comes out that $|h(z)|^{2}\leq 1-|z|^{4}$. And the author says that by using Maximum-Modulus theorem we conclude that $h\equiv 0$. I don't quite understand why is that? If I can prove $h$ is constant I am done, but how Maximum-modulus principle implies that?

Thanks in advance!

Answer 1

Because on $\partial D$ we have $|h(z)|\le 0$ or $|h(z)|=0$ on the boundary. So if the maximum modulus is $0$ then...

Edit, if $h$ is not defined on the boundary then $|h(z)|\le 1-r^4$ for all $z\in B_r(0)$ by maximum modulus. Take limit as $r\to 1$