proof of Schwarz lemma
For any , by the maximal modulus principle must attain its maximum on the closed disk at its boundary , say at some point . But then for any . Taking an infinimum as , we see that values of are bounded: .
Thus . Additionally, , so we see that . This is the first part of the lemma.
Now suppose, as per the premise of the second part of the lemma, that for some . For any , it must be that attains its maximal modulus (1) inside the disk , and it follows that must be constant inside the entire open disk . So for of modulus 1, and , as required.
|Title||proof of Schwarz lemma|
|Date of creation||2013-03-22 12:45:07|
|Last modified on||2013-03-22 12:45:07|
|Last modified by||Mathprof (13753)|