Stable functions and Vietoris' theorem

AbstractAn analytic function f(z) in the unit disc D is called stable if sn(f,·)/f≺1/f holds for all for n∈N0. Here sn stands for the nth partial sum of the Taylor expansion about the origin of f, and ≺ denotes the subordination of analytic functions in D. We prove that (1−z)λ, λ∈[−1,1], are stable. The stability of (1+z)/(1−z) turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials