Meromorphic function which is analytic bounded on $\mathbb H$, has no zeroes on $\mathbb H^-$, modulus 1 on $\mathbb R$

Question

I am interested in the class $G$ of meromorphic function $f:\mathbb C \to \mathbb C$ such that:

1. $f$ is analytic and bounded on the upper half plane $\mathbb H$.
2. $f$ has no zeroes on the lower half plane $\mathbb H^-$.
3. $|f(z)|=1$ for $z\in\mathbb R$.

Such functions form a group under pointwise multiplication of functions. I tried to find some examples on $G$, and obtained: $$f(z)=e^{i\alpha z}, \quad \alpha>0, \qquad f(z) = \frac{z-i\beta^*}{z+i\beta}, \quad \Re \beta>0.$$

Question: Do the above functions generate $G$?

Answer 1

Sketch of the determination of all such functions (a comment that got too long):

Let's move everything to the unit disc by the usual Mobius transform, $z=i\frac{w+1}{1-w}$ where $|w| \le 1$ corresponds to $\Im z >0$ and consider $g(w)=f(z)$ so under the conditions $1,3$ only $|g(w)| \le 1, |w|<1, |g(w)|=1, |w|=1, w \ne 1$.

This means that $g$ is an inner function with singularities only at $1$ hence its singular measure is a positive multiple of the Dirac mass at $1$, while its zeroes may accumulate only at $1$, so it is of the form $B\exp(c\frac{w+1}{w-1}), c >0$ where $B$ is a Blaschke product with possibly infinitely many zeroes accumulating at $1$. Now $B$ is analytic in the whole plane excepting the closure of $\frac{1}{\bar a_n}$, where $a_n$ are its possible finitely or countably many zeroes (with finite multiplicities), so in particular, $B_1(z)=B(w)$ is analytic in the whole plane except with poles at $\bar b_n, b_n=i\frac{a_n+1}{a_n-1}$ (here $1/0$ corresponds to $-i$ under the Mobius transform and as below $i$ has a special role in the Blaschke product if it's infinite)

Since $\exp(c\frac{w+1}{w-1})=\exp{icz}$ and $\frac{w-a}{1-\bar a w}=\frac{z-b}{z-\bar b}$ where $b=i\frac{a+1}{a-1}, \Im b >0$ which of course can be written in the form you wish, but the one here is more customary, and using the standard form for infinite Blaschke products, the full answer is that all functions as required are of the form:

$$f(z)=\gamma e^{ciz}(\frac{z-i}{z+i})^m\Pi_n\frac{|z_n^2+1|}{z_n^2+1}\frac{z-z_n}{z-\bar z_n}$$ where $z_n$ are the zeroes different from $i$, and of course $\Im z_n >0, |z_n| \to \infty, \sum \frac{\Im z_n}{1+|z_n|^2}< \infty$ if there are infinitely many such, while $|\gamma|=1, c>0, m \ge 0$ (since $0$ gets special treatment in the infinite Blaschke products in the plane, so $i$ as its image does in the upper plane - if the product is finite, this doesn't matter as one can absorb the constants in the product in front)