Exercise 7. The family of mappings introduced here plays an important role in complex analysis. These mappings, sometimes called Blaschke factors, will reappear in various applications in later chapters.
(1) Let z, w be two complex numbers such that zw 6= 1. Prove that w − z 1 − wz < 1 if |z| < 1 and |w| < 1, and also that w − z 1 − wz = 1 if |z| = 1 or |w| = 1.
(2) Prove that for a fixed w in the unit disc D, the mapping F : z 7→ w − z 1 − wz satisfies the following conditions
(a) F maps the unit disc to itself (that is, F : D → D), and is holomorphic.
(b) F interchanges 0 and w, namely F(0) = w and F(w) = 0.
(c) |F(z)| = 1 if |z| = 1.
(d) F : D → D is bijective.
Why Choose Us
We value our clients. For this reason, we ensure that each paper is written carefully as per the instructions provided by the client. Our editing team also checks all the papers to ensure that they have been completed as per the expectations.
Professional Academic Writers
Over the years, our Acme Homework has managed to secure the most qualified, reliable and experienced team of writers. The company has also ensured continued training and development of the team members to ensure that it keep up with the rising Academic Trends.
Our prices are fairly priced in such a way that ensures affordability. Additionally, you can get a free price quotation by clicking on the "Place Order" button.
We pay strict attention on deadlines. For this reason, we ensure that all papers are submitted earlier, even before the deadline indicated by the customer. For this reason, the client can go through the work and review everything.
At Essay USA, all papers are plagiarism-free as they are written from scratch. We have taken strict measures to ensure that there is no similarity on all papers and that citations are included as per the standards set.
Customer Support 24/7
Our support team is readily available to provide any guidance/help on our platform at any time of the day/night. Feel free to contact us via the Chat window or support email: email@example.com.
Try it now!
How it works?
Follow these simple steps to get your paper done
Place your order
Fill in the order form and provide all details of your assignment.
Proceed with the payment
Choose the payment system that suits you most.
Receive the final file
Once your paper is ready, we will email it to you.
Essay USA has stood as the world’s leading custom essay writing services providers. Once you enter all the details in the order form under the place order button, the rest is up to us.
At Essay USA, we prioritize on all aspects that bring about a good grade such as impeccable grammar, proper structure, zero-plagiarism and conformance to guidelines. Our experienced team of writers will help you completed your essays and other assignments.
Admission and Business Papers
Be assured that you’ll definitely get accepted to the Master’s level program at any university once you enter all the details in the order form. We won’t leave you here; we will also help you secure a good position in your aspired workplace by creating an outstanding resume or portfolio once you place an order.
Editing and Proofreading
Our skilled editing and writing team will help you restructure you paper, paraphrase, correct grammar and replace plagiarized sections on your paper just on time. The service is geared toward eliminating any mistakes and rather enhancing better quality.
We have writers in almost all fields including the most technical fields. You don’t have to worry about the complexity of your paper. Simply enter as much details as possible in the place order section.