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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: {\it R. Askey} and {\it 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); {\it St. Ruscheweyh}, SIAM, J. Math. Anal. 9, 682--686 (1978; Zbl 0391.30010)]. In [{\it St. Ruscheweyh} and {\it L. Salinas}, Ann. Univ. Mariae Curie-Skł odowska, Sect. A 54, 117--123 (2000; Zbl 0989.30010)], the authors presented a new notion for holomorphic functions $f(z)$ in the unit disk $\Bbb{D}=\{z: |z|<1\}.$ Namely, a holomorphic function in $\Bbb{D}$ is stable if $\frac{s_n(f)}{f}\prec\frac{1}{f}$ holds for all $n=0, 1, 2, \ldots .$ \newline 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 $\Bbb{D}.$ 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.

30C10Polynomials (one complex variable)
30C15Zeros of polynomials, etc. (one complex variable)
26C05Polynomials: analytic properties (real variables)
33C45Orthogonal polynomials and functions of hypergeometric type
Full Text: DOI
[1] Askey, R.: Orthogonal polynomials and special function. Regional conf. Lect. appl. Math. 21 (1975) · Zbl 0298.33008
[2] Askey, R.; Steinig, J.: Some positive trigonometric sums. Trans. amer. Math. soc. 187, 295-307 (1974) · Zbl 0244.42002
[3] Ruscheweyh, St: On the Kakeya--Eneström theorem and Gegenbauer polynomial sums. SIAM J. Math. anal. 9, 682-686 (1978) · Zbl 0391.30010
[4] Ruscheweyh, St; Salinas, L.: On starlike functions of order ${\lambda}$\in[1/2,1. Ann. univ. Mariae Curie-skłodwska 54, 117-123 (2000)
[5] Vietoris, L.: Über das vorzeichen gewisser trigonometrischer summen. Sitzungsber. österr. Akad. wiss. 167, 125-135 (1958) · Zbl 0088.27402
[6] Wolfram Research, Inc., Mathematica, Version 4.1, Champaign, IL, 1999