Holomorphic function on a unit disk

Question

Let $f$ be a holomorphic function on the unit disk. If $|f(z)|=1$, show that $f$ must be a constant function.

I am a little confused how to start this.

Answer 1

Let $f(z)=u(x,y)+\text{i}v(x,y),$ where $z=x+\text{i}y.$

As $|f(z)|=1,$ we know that $u^2+v^2=1.$ This implies that $$uu_x+vv_x=0,uu_y+vv_y=0.$$

So by the Cauchy-Riemann equations, $$uu_x-vu_y=0,uu_y+vu_x=0.$$ Eliminating $u_y$ gives $0=(u^2+v^2)=u_x.$ So $u_x=0$ everywhere in the unit disc. We can conclude the same for $u_y,v_x,\text{ and }v_y.$

Then $f'(z)=0$ for all $z$ in the unit disc, so $f$ is constant.