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On geometric estimates for univalent functions

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Abstract

We provide geometric estimates for conformal maps of the unit disc. As a consequence, a bound for the derivative of such maps follows yielding, in particular, an answer to a question posed by Martín Chuaqui. Finally, estimates regarding the curvature of image arcs under conformal maps are also obtained.

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Acknowledgements

The authors would like to thank the referee for a careful reading of the manuscript.

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Authors and Affiliations

  1. Departamento de Análisis Matemático y Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias No 3, 28040, Madrid, Spain

    Eva A. Gallardo-Gutiérrez

  2. Instituto de Ciencias Matemáticas ICMAT (CSIC-UAM-UC3M-UCM), C/ Nicolás Cabrera, 13-15, 28049, Madrid, Spain

    Eva A. Gallardo-Gutiérrez

  3. Institut für Mathematik, MA 8-2, Technische Universität, 10623, Berlin, Germany

    Christian Pommerenke

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  1. Eva A. Gallardo-Gutiérrez
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  2. Christian Pommerenke
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Correspondence to Eva A. Gallardo-Gutiérrez.

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E. A. Gallardo-Gutiérrez is partially supported by Plan Nacional I+D Grants No. MTM2016-77710-P and PID2019-105979GB-I00, Spain, Grupo UCM 910346, the Spanish Ministry of Science and Innovation, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001)

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Gallardo-Gutiérrez, E.A., Pommerenke, C. On geometric estimates for univalent functions. Annali di Matematica 200, 1075–1084 (2021). https://doi.org/10.1007/s10231-020-01026-w

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  • DOI: https://doi.org/10.1007/s10231-020-01026-w

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