Stable functions and Vietoris' theorem
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.
scientific article; zbMATH DE number 2075645