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Blaschke Products and Circumscribed Conics

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Abstract

We study geometrical properties of finite Blaschke products. For a Blaschke product B of degree d, let \(L_{\lambda }\) be the set of the lines tangent to the unit circle at the d preimages \( B^{-1}(\lambda ) \). We show that the trace of the intersection points of each pair of two elements in \( L_{\lambda } \) as \( \lambda \) ranges over the unit circle forms an algebraic curve of degree at most \( d-1 \). In case of low degree, we have more precise results. For instance, for \( d=3 \), the trace forms a conic section. For \( d=4 \), we provide a necessary and sufficient condition for Blaschke products whose trace include a conic section.

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Acknowledgements

I would like to thank Professor Masahiko Taniguchi for useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, National Defense Academy, Yokosuka, 239-8686, Japan

    Masayo Fujimura

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  1. Masayo Fujimura
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Correspondence to Masayo Fujimura.

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Communicated by Kenneth Stephenson.

This work was partially supported by JSPS KAKENHI Grant Number JP15K04943.

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Fujimura, M. Blaschke Products and Circumscribed Conics. Comput. Methods Funct. Theory 17, 635–652 (2017). https://doi.org/10.1007/s40315-017-0201-7

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  • DOI: https://doi.org/10.1007/s40315-017-0201-7

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