modulus-related analytic functions

Question

Let $D\subset\mathbb{C}$ be the open unit disk. Suppose $f,g,F,G:D\rightarrow\mathbb{C}$ are analytic functions linked by $$\vert f(z)\vert^2+\vert g(z)\vert^2=\vert F(z)\vert^2+\vert G(z)\vert^2; \qquad \forall z\in D.$$

Question 1. If $f\neq \alpha g$ and $g\neq\beta f$ for any $\alpha, \beta\in\mathbb{C}$ then is the same true for $F$ and $G$?

Caveat. Not true for real analytic functions: take $f=\sin x, \,g=\cos x, \, F=\frac1{\sqrt{2}}=G$.

EDIT. Encouraged by Christian's positive answer, let's upgrade the problem.

Question 2. Suppose $f, g, h$ are linearly independent (over $\mathbb{C}$) such that $$\vert f\vert^2+\vert g\vert^2+\vert h\vert^2=\vert F\vert^2+\vert G\vert^2+\vert H\vert^2.$$ Should $F, G, H$ be linearly independent, too?

Answer 1

In fact, more is true. If $f_j$ are linearly independent and $F_i$ are linearly independent, and $$\sum_{j=1}^n|f_j|^2=\sum_{i=1}^m|F_i|^2,$$ then $m=n$ and $F_i$ are obtained from $g_i$ by a unitary transformation. See, for example, https://arxiv.org/pdf/math/0007030.pdf, section 3. This is called the "Calabi rigidity".

Answer 2

Yes, for Question 1.

We can divide through by one of the functions on the left-hand side and also assume $F=G$, and then the question becomes: Can $1+|h(z)|^2$ be the square of the absolute value of a holomorphic function, for a non-constant holomorphic $h$?

For this to happen, we need $\log (1+|h(z)|^2)$ to be harmonic. If you now just work out the Laplacian of this function and use the Cauchy-Riemann equations for $h$, you'll find that the numerator of the expression you obtain from two applications of the quotient rule equals $4((h_1)_x^2+(h_2)_x^2)$, so $h$ has to be constant.