Computational Methods and Function Theory

, Volume 15, Issue 2, pp 323–371

# Level Curve Configurations and Conformal Equivalence of Meromorphic Functions

• Trevor Richards
Article

## Abstract

Let $$f=B_1/B_2$$ be a ratio of finite Blaschke products having no critical points on $$\partial \mathbb {D}$$. Then $$f$$ has finitely many critical level curves (level curves containing critical points of $$f$$) 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 $$f$$, one needs only understand the configuration of the finitely many critical level curves of $$f$$. In this paper, we show that in fact such a function $$f$$ is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if $$f_1$$ and $$f_2$$ have the same configuration of critical level curves, then there is a conformal map $$\phi$$ such that $$f_1\equiv f_2\circ \phi$$. 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.

## Keywords

Complex analysis Meromorphic functions Level curves  Critical points Critical values

Primary 30C 30A

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