Access the full text.
Sign up today, get DeepDyve free for 14 days.
Let $$f=B_1/B_2$$ f = B 1 / B 2 be a ratio of finite Blaschke products having no critical points on $$\partial \mathbb {D}$$ ∂ D . Then $$f$$ f has finitely many critical level curves (level curves containing critical points of $$f$$ f ) in the disk, and the non-critical level curves interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of $$f$$ f , one needs only understand the configuration of the finitely many critical level curves of $$f$$ f . In this paper, we show that in fact such a function $$f$$ f is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if $$f_1$$ f 1 and $$f_2$$ f 2 have the same configuration of critical level curves, then there is a conformal map $$\phi $$ ϕ such that $$f_1\equiv f_2\circ \phi $$ f 1 ≡ f 2 ∘ ϕ . We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss–Lucas theorem to rational functions) using level curves.
Computational Methods and Function Theory – Springer Journals
Published: Mar 31, 2015
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.