Level Sets of the Hyperbolic Derivative for Analytic Self-Maps of the Unit Disk

Abstract

Let the function \(\varphi \) be holomorphic in the unit disk \({\mathbb {D}}\) of the complex plane \({\mathbb {C}}\) and let \(\varphi ({\mathbb {D}})\subset {\mathbb {D}}\). We study the level sets and the critical points of the hyperbolic derivative of \(\varphi \),

$$\begin{aligned} |D_{\varphi }(z)|:=\frac{(1-|z|^2)|\varphi '(z)|}{1-|\varphi (z)|^2}. \end{aligned}$$

In particular, we show how the Schwarzian derivative of \(\varphi \) reveals the nature of the critical points.

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Acknowledgements

The authors want to thank the anonymous referee for his/her careful reading and comments that improved the clarity of this paper, and helped us to correct some errors. Also the authors want to thank Fernando Morales, our colleague at the math department of our university, who helped us greatly with the figures of examples in Sect. 4. Finally, the authors wish to thank Universidad Nacional de Colombia, Sede Medellín, for supporting this work through Project Hermes 49148.

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Correspondence to Diego Mejía.

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Communicated by Oliver Roth.

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Arango, J., Arbeláez, H. & Mejía, D. Level Sets of the Hyperbolic Derivative for Analytic Self-Maps of the Unit Disk. Complex Anal. Oper. Theory 15, 31 (2021). https://doi.org/10.1007/s11785-021-01077-8

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Keywords

  • Hyperbolic derivative
  • Level set
  • Critical point
  • Trajectory
  • Schwarzian derivative
  • Hyperbolic order

Mathematics Subject Classification

  • 30H05
  • 30E99
  • 30J99