I am trying to prove the claim as the title. Here by involution map I mean a map after composing itself becomes identity. So if such map restricts to the boundary of the disk, it turns out that this map must be identity on the whole disk.
I only observe that such map always swap points, i.e. if $f(a)=b$, then $f(b)=a$. But any way, I have no idea of how to relate this to the condition that $f$ is identity on the boundary. Any idea? Thanks!