Stable functions and Vietoris' theorem

An 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