proof of Schwarz lemma

Define g(z)=f(z)/z. Then g:Δ is a holomorphic functionMathworldPlanetmath. The Schwarz lemmaMathworldPlanetmath is just an application of the maximal modulus principle to g.

For any 1>ϵ>0, by the maximal modulus principle |g| must attain its maximum on the closed disk {z:|z|1-ϵ} at its boundary {z:|z|=1-ϵ}, say at some point zϵ. But then |g(z)||g(zϵ)|11-ϵ for any |z|1-ϵ. Taking an infinimum as ϵ0, we see that values of g are bounded: |g(z)|1.

Thus |f(z)||z|. Additionally, f(0)=g(0), so we see that |f(0)|=|g(0)|1. This is the first part of the lemma.

Now suppose, as per the premise of the second part of the lemma, that |g(w)|=1 for some wΔ. For any r>|w|, it must be that |g| attains its maximal modulus (1) inside the disk {z:|z|r}, and it follows that g must be constant inside the entire open disk Δ. So g(z)a for a=g(w) of modulus 1, and f(z)=az, as required.

Title proof of Schwarz lemma
Canonical name ProofOfSchwarzLemma
Date of creation 2013-03-22 12:45:07
Last modified on 2013-03-22 12:45:07
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 6
Author Mathprof (13753)
Entry type Proof
Classification msc 30C80