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Stable functions and vietoris’ theorem. (English) Zbl 1052.30011

Positivity results for trigonometric sums and their generalizations to some polynomial inequalities have long history and important applications [see for example the following papers: R. Askey and G. Gasper, Inequalities for polynomials, in The Bieberbach Conjecture: Proc. of the Symposium on the Ocassion of the Proof, A. Baernstein, D. Drasin, P. Duren and A. Marden, eds., (Math. Surveys Monographs 21, American Mathematical Society, Providence, R. I.), 7–32 (1986; Zbl 0631.30001); Am. J. Math., 98, 709–737 (1976; Zbl 0355.33005); St. Ruscheweyh, SIAM, J. Math. Anal. 9, 682–686 (1978; Zbl 0391.30010)].

In [St. Ruscheweyh and 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 $f\left(z\right)$ in the unit disk $𝔻=\left\{z:|z|<1\right\}·$ Namely, a holomorphic function in $𝔻$ is stable if $\frac{{s}_{n}\left(f\right)}{f}\prec \frac{1}{f}$ holds for all $n=0,1,2,...·$

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 $𝔻·$ Proving that some particular functions are stable (e.g. ${\left(1-z\right)}^{\lambda },\lambda \in \left[-1,1\right]$) 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.

##### MSC:
 30C10 Polynomials (one complex variable) 30C15 Zeros of polynomials, etc. (one complex variable) 26C05 Polynomials: analytic properties (real variables) 33C45 Orthogonal polynomials and functions of hypergeometric type