I have the following problem:
Let $U_1,U_2\subset\mathbb{C}$ with $\pi_1(U_i)=0$ and take $a_i\in U_i.$ Let $f:U_1\to U_2$ be a biholomorphism with $f(a_1)=a_2$ and let $g:U_1 \to U_2$ be holomorphic and injective with $g(a_1)=a_2.$ Show that $|g'(a_1)|\leq |f'(a_1)|.$
I know that if $g$ is holomorphic and injective, $g'(z)\neq0$ for all $z\in U_1.$ I don't know how to use the fact that $U_i$ are simply connected. I've tried solving this using Cauchy's integral formula, I've tried doing some algebraic gymnastics with the theorem for differentiating the inverse of a holomorphic function and nothing. Also this is not true on $\mathbb{R}$ and for differentiable functions. Take $f(x)=x,$ $g(x)=x^3,$ $a_1=1$ and $a_2=1$ and take $U_1$ in such a way so that it contains the origin.