Question about Christoffel-Schwarz mappings from Ahlfors

1. Apr 21, 2010

zpconn

Suppose $$\zeta$$ lies in the upper half $$S_k$$ of an origin-centered disc in the complex plane and consider the function $$g(\zeta) = f(\zeta^{\alpha_k} + z_k)$$ where $$0 < \alpha_k < 2$$ is a real constant, $$z_k$$ is a complex constant, and $$f$$ is a bijective conformal map from its domain (which includes the points $$\zeta^{\alpha_k} + z_k$$) to the open unit disc. (For the meanings of these constants, see Pg. 235 of Ahlfors if you have access to it.)

The reflection principle of Schwarz guarantees that $$g$$ has an analytic continuation to the full disc, and hence it is analytic at the origin and has the Taylor expansion

$$f(z_k + \zeta^{\alpha_k}) = f(z_k) + \sum_{m = 1}^{\infty} a_m \zeta^m.$$

This is all fine, but now Ahlfors asserts that $$a_1$$ is nonzero because if it were zero then the half-disc $$S_k$$ would not be mapped into the unit disc by $$g$$.

I'm not seeing why. But this is a crucial step. Any ideas?