Stable functions and Vietoris' theorem
type: Item
instance of: scholarly article
title: Stable functions and Vietoris' theorem
zbMATH Open document ID: 1052.30011
author: Stephan Ruscheweyh
author: Luis Salinas
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-06-15T00:00:00Z
review text: Positivity results for trigonometric sums and their generalizations to some polynomial inequalities have long history and important applications [see for example the following papers: \textit{R. Askey} and \textit{G. Gasper}, Inequalities for polynomials. Math. Surv. Monogr. 21, 7--32 (1986; Zbl 0631.30001); Am. J. Math. 98, 709--737 (1976; Zbl 0355.33005); \textit{St. Ruscheweyh}, SIAM, J. Math. Anal. 9, 682--686 (1978; Zbl 0391.30010)]. In [\textit{St. Ruscheweyh} and \textit{L. Salinas}, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 54, 117--123 (2000; Zbl 0989.30010)], the authors presented a new notion for holomorphic functions \(f(z)\) in the unit disk \(\mathbb{D}=\{z: |z|<1\}\). Namely, a holomorphic function in \(\mathbb{D}\) is stable if \(\frac{s_n(f)}{f}\prec\frac{1}{f}\) holds for all \(n=0, 1, 2, \ldots\). Here \(s_n\) denotes the \(n\)-th partial sum of the Taylor expansion about the origin, and \(\prec\) denotes the subordination of holomorphic functions in \(\mathbb{D}\). Proving that some particular functions are stable (e.g. \((1-z)^{\lambda}, \lambda \in [-1, 1]\)) the authors were able to transform this property into some old and new trigonometric inequalities. The method of stable functions appears to be pretty powerful, which has been discussed by the authors as well, by considering some conjectures and their validity and applications.
reviewed by: Jan Szynal
Mathematics Subject Classification ID: 30C10
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 2075645
zbMATH Keywords: subordination
zbMATH Keywords: stable functions Gegenbauer polynomials
describes a project that uses: Mathematica
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.11.035
OpenAlex ID: W2003940436
cites work: Q4052481
cites work: Some Positive Trigonometric Sums
cites work: On the Kakeya–Eneström Theorem and Gegenbauer Polynomial Sums
cites work: On starlike functions of order \(\lambda\in[{1\over 2},1)\)
cites work: Q3261493
DOI: 10.1016/J.JMAA.2003.11.035
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description: scientific article; zbMATH DE number 2075645