The Schwarz Lemma -- Complex Analysis

If you're interested in personal help, I've posted my tutoring services on Fiverr: www.fiverr.com...
I have not had the opportunity to teach mathematics as much lately, given the amount of focus I have given to my research. I enjoy the process of teaching and interacting with students and so have decided to do some online tutoring.
Part 1 -- The Maximum Principle: • MAXIMUM PRINCIPLE -- P...
Part 3 -- Liouville's theorem: • Liouville's theorem --...
In today's video, we want to take a look at the Schwarz lemma - this is a monumental result in the subject of one complex variable, and has lead to many developments in various areas of mathematics. The so-called Ahlfors-Schwarz lemma, and the Yau-Schwarz lemma are extremely powerful tools in the area of complex differential geometry, and are complex analogues of the Laplacian of the energy density in the study of harmonic maps. The Schwarz lemma is also a motivating result and the initial kindling for the study of Kobayashi hyperbolic manifolds, which are objects of great interest today.
The Schwarz lemma describes the behaviour of holomorphic maps from the unit disk to itself. It says that if f is a holomorphic map from the unit disk to itself, then it is distance decreasing. On the face of it, this does not sound too interesting, but it turns out to have powerful applications. In particular, the Schwarz lemma can be used to give a proof of the famous Liouville theorem, which states that a bounded entire function is constant.
👍 To support the channel, hit the like button and subscribe.
🐸 To learn how I make these animations using Grant Sanderson's (3Blue1Brown) program Manim, check out my quick intro to Manim on Skillshare: www.skillshare...
🎸 Music in this video is from Epidemic Sound: www.epidemicso...
Ways to support the channel:
🏆 Support the channel over on Patreon / kylebroder
🐸 Support the channel by signing up to a free trial of Skillshare using the link www.skillshare...
FYI, these are affiliate links 💪🙏
These videos are separate from my research and teaching roles at the Australian National University, University of Sydney, and Beijing University.

[@guilhermefranco2949]

Also, by proving that |g(z)|

[@MrAAMNNITAllahabad]

How come r tending to be 1?? Please, let me know.

[@changtillend5718]

Thank you ...

[@akshatjangra4167]

Why do you add that these videos are independent of your research?

[@gavincourts854]

probably legal reasons

[@akshatjangra4167]

@@gavincourts854 yah probably

[@KyleBroder]

Yeah

[@jamesking8043]

One down! 136 to go!

[@visualgebra]

Wow!

[@KyleBroder]

💪

[@Atseror]

it’s cool to copy the demonstration, but this video lacks intuitive explaination

[@KyleBroder]

Thanks for the feedback