Help on a problem from Ahlfors' Complex Analysis

Question

This is a problem on Ahlfors' Complex Analysis.

If $g(w)$ and $f(z)$ are analytic functions, show that $g(f(z))$ is also analytic.

I've tried this and I have stumbled upon a tricky partial differential equation. I can't seem to use the Cauchy-Riemann condition of $g, f$ and use it to get the result. Any help will be appreciated.

Answer 1

In my copy of Rudin's Principles of Mathematical Analysis the real cain rule is Theorem $5.5$ on page 90. Exactly the same proof can be used to show that when $f$ is complex differentiable at $z_0$ and $g$ is complex differentiable at $w_0=f(z_0)$ then $h:=g\circ f$ is complex differentiable at $z_0$. By complex differentiable I mean that $$\lim_{z\to z_0}{f(z)-f(z_0)\over z-z_0}\in{\mathbb C}$$ exists.

Answer 2

The best way to do this is to use the chain rule phrased in terms of $\partial/\partial z$ and $\partial/\partial\bar z$. See my answer here. Conveniently, Ahlfors defines those on the page before your question appears.