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Geometric Aspects of Functional Analysis pp 123–136Cite as

\(\boldsymbol{(s,p)}\)-Valent Functions

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2169))

Abstract

We introduce the notion of \((\mathcal{F},p)\)-valent functions. We concentrate in our investigation on the case, where \(\mathcal{F}\) is the class of polynomials of degree at most s. These functions, which we call (s, p)-valent functions, provide a natural generalization of p-valent functions (see Hayman, Multivalent Functions, 2nd ed, Cambridge Tracts in Mathematics, vol 110, 1994). We provide a rather accurate characterizing of (s, p)-valent functions in terms of their Taylor coefficients, through “Taylor domination”, and through linear non-stationary recurrences with uniformly bounded coefficients. We prove a “distortion theorem” for such functions, comparing them with polynomials sharing their zeroes, and obtain an essentially sharp Remez-type inequality in the spirit of Yomdin (Isr J Math 186:45–60, 2011) for complex polynomials of one variable. Finally, based on these results, we present a Remez-type inequality for (s, p)-valent functions.

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Author information

Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75005, Paris, France

    Omer Friedland

  2. Department of Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Yosef Yomdin

Authors
  1. Omer Friedland
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  2. Yosef Yomdin
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Correspondence to Omer Friedland .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel-Aviv, Israel

    Bo'az Klartag

  2. Mathematics Department, Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Friedland, O., Yomdin, Y. (2017). \(\boldsymbol{(s,p)}\)-Valent Functions. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2169. Springer, Cham. https://doi.org/10.1007/978-3-319-45282-1_8

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