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Smale’s mean value conjecture for finite Blaschke products

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Abstract

Motivated by a dictionary between polynomials and finite Blaschke products, we study both Smale’s mean value conjecture and its dual conjecture for finite Blaschke products in this paper. Our result on the dual conjecture for finite Blaschke products allows us to improve a bound obtained by Dubinin and Sugawa for the dual mean value conjecture for polynomials.

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Acknowledgments

The authors want to thank the referee for her/his many very helpful suggestions, in particular, a very detailed suggestion of a more topological proof of Theorem 3. We have followed the suggestion of making use of the topological theory of covering spaces as well as applying a better estimate of \(\mu (s)\), the modulus of the extreme Grötzsch ring \(\mathbb {D}-[0,s]\) in Sect. 4. We also thank Toshiyuki Sugawa and Yum Tong Siu for the suggestion of the connection between polynomials and finite Blaschke products through some rescalings.

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Authors and Affiliations

  1. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, China

    Tuen-Wai Ng & Yongquan Zhang

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  1. Tuen-Wai Ng
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  2. Yongquan Zhang
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Correspondence to Tuen-Wai Ng.

Additional information

Tuen-Wai Ng: Partially supported by RGC grant HKU 704611P and HKU 703313P.

Yongquan Zhang: Partially supported by a summer research fellowship of Faculty of Science, HKU. Dedicated to David Minda on the occasion of his retirement.

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Ng, TW., Zhang, Y. Smale’s mean value conjecture for finite Blaschke products. J Anal 24, 331–345 (2016). https://doi.org/10.1007/s41478-016-0007-4

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  • DOI: https://doi.org/10.1007/s41478-016-0007-4

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