Suppose $f_1, f_2$ are holomorphic function on the disk unit $\mathbb D.$ If $\left|f_1\left(z\right)\right|+\left|f_2\left(z\right)\right|=1, \forall z\in \mathbb D.$ Prove that $f_1$ and $f_2$ are constants.
My attempt $\left|f_1\left(z\right)\right|+\left|f_2\left(z\right)\right|=1, \forall z\in \mathbb D.$ $\implies$ $|f_1(z)|<1$ and $|f_2(z)|<1$, If the domain is $\mathbb C$ instead of $\mathbb D$, I could I have apply lioville's theorem.
Since, $\left|f_1\left(z\right)\right|+\left|f_2\left(z\right)\right|=1$, so $f_1(0)=0$ and $f_2(0)=0$ is not true simultaneously. So, I can not use Shwartz lemma. I am helpless. Please help me.
