This is a follow-up to Looking for holomorphic function on a sector with specified boundary behavior. There I was looking for a holomorphic function on a sector with real boundary condition on one boundary ray and a condition on its norm on the other boundary ray. Alexandre Eremenko gave a complete and very nice answer to that question. This time, I'm looking for a similar function but where there is now a non-local boundary condition on its norm.

Fix $h \in (0,\pi/2)$. I am trying to explicitly exhibit a nonconstant holomorphic function $f: \Sigma \to \mathbb{C}$, where $\Sigma$ is the punctured first quadrant

$$\Sigma := \{z \in \mathbb{C} \:|\: z \neq 0, 0 \leq \arg z \leq \pi/2\},$$

and where we require f to satisfy the following conditions on $\partial\Sigma$:

• $f(x) \in \mathbb{R}$ for $x \in \mathbb{R}_+$.

• $2|f(ix)|^2 = |f(e^{ih}x)|^2+1$ for $x \in \mathbb{R}_+$.

In the linked question, any two satisfactory functions were related by multiplication by a Blaschke product; in this case, there does not seem to be a similar way to go from knowing one satisfactory function to knowing all of them.