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Published in:
2017 | OriginalPaper | Chapter
\(\boldsymbol{(s,p)}\)-Valent Functions
Authors : Omer Friedland, Yosef Yomdin
Published in: Geometric Aspects of Functional Analysis
Publisher: Springer International Publishing
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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.