Let $f(z)$ be a function on the unit disk $\mathbb{D}$ which is meromorphic on $\mathbb{D}$ with only one simple pole at $z = 1/2$ Let $f(z)$ be a function on the unit disk $\mathbb{D}$ which is meromorphic on $\mathbb{D}$ with only one simple pole at $z = 1/2$ and which is continuous up
to $∂\mathbb{D}$ and $|f(z)| ≡ 1$ along $∂\mathbb{D}.$ Write down all functions described by the above.
This is one over several parts of a question from an old qual.  I want to say there are no such functions, but I do not know how to go about this.  And suggestions?  Thanks.  I tried using shwartz?
 A: First, we discuss the zeros of $f(z)$. Because we have continuity on the boundary, there can be no accumulation point of zeros at the boundary. Next, because $1/2$ is a simple pole there can be no accumulation point of zeros at $1/2$ either. Thus we conclude that $f(z)$ must have a finite number of zeros (since if they accumulated to a point strictly in $\mathbb{D}-\{1/2\}$, then $f(z)$ would be the zero function). Let $$z_1,\ldots,z_n$$ be the zeros of $f(z)$. 
We now show that there exists an $\alpha\in\mathbb{C}$, $|\alpha|=1$, such that $$f(z)=\alpha\cdot \frac{z-2}{1-2z}\cdot\frac{z-z_1}{1-\overline{z_1}z}\cdots\frac{z-z_n}{z-\overline{z_n}z}. $$ 
Proof. Consider the function, $$g(z):=f(z)\frac{1-2z}{z-2}\cdot\frac{1-\overline{z_1}z}{z-z_1}\cdots\frac{1-\overline{z_n}z}{z-z_n}.$$ We notice, several important facts, $g(z)$ is continuous on $\partial \mathbb{D}$, with $|g(z)|=1$ for $|z|=1$, $g(z)$ is analytic on all of $\mathbb{D}$ (we removed the one singularity that $f(z)$ had), and most importantly, $g(z)$ is not zero in all of $\mathbb{D}$. Hence, neither $g(z)$ nor $\frac{1}{g(z)}$ achieve a maximum on $\mathbb{D}$ (maximum modulus principle). Thus, $g(z)$ must be a constant. But by continuity on the boundry, $|g(z)|=1$, hence there is an $|\alpha|=1$, such that $g(z)=\alpha$, which is what we wished to show. 
In general, $$\frac{z-w}{1-\overline{w}z}$$ are called Blaschke products. They typically are used to categorize functions that map the unit circle to the unit circle, since they themselves map the unit circle to the unit circle. 
