TITLE: Blaschke Factor mapping holomorphic
QUESTION [7 upvotes]: In Stein and Shakarchi's book, Chapter 1, Exercise 7 asks us to show that $$\left|\frac{w-z}{1-\overline{w}{z}}\right|<1$$ if $|z|<1$ and $|w|<1$, with equality if either $|z|=1$ or $|w|=1$. I was able to show this much, and for the second part of the question, most of it is immediate from the above, except showing that for a fixed $w\in\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ $$F(z)=\frac{w-z}{1-\overline{w}{z}}$$ is holomorphic. I can do it if I completely expand everything and get it to the form $F(x,y)=u(x,y)+iv(x,y)$ (I haven't actually written everything out, only assured myself that I could actually get it to such a form). From here I should be able to check the Cauchy-Riemann equations and verify the partial derivatives are continuous, which is enough to show that $F$ is holomorphic, but I was wondering if there was a better/more elegant way of showing it? Thanks!

REPLY [9 votes]: Quotients of holomorphic functions are holomorphic where they're defined -- that is, wherever the denominator is nonzero.
In your case, the denominator is never zero because.... (details left to you)
Edit: Of course, you do have to mention (or prove) that both the numerator and denominator are, in fact, holomorphic!
And if this fact about quotients hasn't been proven... well, I guess you'll have to do that, too.  The proof is exactly the same as the real-variable case.