Given 2 sequences $(a_n)_{n\in\mathbb{Z}}$, $(b_n)_{n\in\mathbb{Z}}$ with $b_n<a_n<a_{n+1}$, and $a_{-1}<0<b_1$ I'm trying to show that the function defined by
$$\theta(z)=\frac{a_0-z}{b_0-z}\prod_{n\in\mathbb{Z}\backslash\{0\}}\frac{1-\frac{z}{a_n}}{1-\frac{z}{b_n}}$$
Maps the upper half plane, $\mathbb{H}$ to itself. So far I have shown that the product converges, and that the function is holomorphic in the upper half plane, and that it is meromorphic in $\mathbb{C}$ with all zeros and all poles being simple and on the real line (the zeros being the $a_n$'s and the poles being the $b_n$'s).
I've also noticed that if $z$ is in the upper half-plane, then $\frac{1-\frac{z}{a_n}}{1-\frac{z}{b_n}}$ is also in the upper half-plane when $n>0$ and in the lower half-plane when $n<0$, and the factor with $a_0$ and $b_0$ is also in the upper half-plane, when $z$ is.
My main idea is to show that the partial products map $\mathbb{H}$ to $\mathbb{H}$ (at least for N large enough), so that the limit function, $\theta$, maps $\mathbb{H}$ to $\{z|\text{Im}z\geq0\}$, and then I'd use the Open Mapping Theorem to establish that the limit indeed maps $\mathbb{H}$ to $\mathbb{H}$. This hasn't been succesful so far. I've also tried looking at a holomorphic logarithm in $\mathbb{H}$ and looking at the logarithmic derivative, but this hasn't worked either.
I'd really appreciate a hint or something pointing me in a direction that could be fruitful. Thanks!